After four weeks of terror and retaliation in Israel and Gaza, and 20 months of war in Ukraine, President Biden is confronting the limits of his leverage in the two international conflicts defining his presidency.

For 10 days, the Biden administration has been urging Prime Minister Benjamin Netanyahu to allow for “humanitarian pauses” in the bombing of Gaza, hoping that the $3.8 billion a year in American security assistance would carry with it enough influence over the Israeli leader’s tactics.

It has not. Mr. Netanyahu rebuffed Mr. Biden’s push for greater efforts to avoid civilian casualties in a phone call on Monday. And he has pushed ahead with what he has called “mighty vengeance” for the Oct. 7 attacks, using huge bombs to collapse Hamas’s network of tunnels, even if they also collapse whole neighborhoods in Gaza.

In Ukraine, the country’s most senior military commander, Gen. Valery Zaluzhny, uttered the word last week that American officials carefully avoided for the better part of a year: stalemate. Many of Mr. Biden’s aides agree that Ukraine and Russia are dug in, unable to move the front lines of the battle in any significant way.

But they fear that General Zaluzhny’s candor will make it harder to get Republicans to vote for aggressive funding for the war — and may encourage President Vladimir V. Putin of Russia to dig in, hoping former President Donald J. Trump or a Republican with similar views will be elected next year and pull back American support.

In both cases, Mr. Biden’s influence over how his allies prosecute those wars seems far more constrained than expected, given his central role as the supplier of arms and intelligence. But because the United States is so tied to both struggles, as Israel’s most powerful ally and Ukraine’s best hope of remaining a free and independent nation, the president’s legacy is tied to how those countries act, and how the wars end.

“There is a long history of U.S. presidents realizing they don’t have as much leverage over Israel as they thought,” said Representative Seth Moulton, a Massachusetts Democrat and former Marine who served four tours in Iraq. And he said the same applies to Ukraine, “where this is first and foremost their fight, even if we have huge stakes in the outcome.”

History, geography and American national interests separate these two radically different conflicts, though it was Mr. Biden himself who joined them in a speech to the nation two weeks ago after returning from a visit to Israel, where he mourned the loss of 1,400 people in the Oct. 7 attacks and vowed to join in the dismantling of Hamas.

“Hamas and Putin represent different threats,” he said that evening, “but they share this in common: They both want to completely annihilate a neighboring democracy — completely annihilate it.”

Mr. Biden brings a passion to these two fights that reaches back to his days as chairman of the Senate Foreign Relations Committee and as vice president. His aides report that he believes history will remember him for how he defended democracy against forces of chaos, terror and dictatorship.

At the same time, the president is a cautious player, and in both conflicts he has repeatedly said American forces will not directly enter the battle, as long as Americans in the Middle East or NATO nations are not the subject of sustained attack. He entered politics when America was deep in the Vietnam War, a searing experience for him, and he spent much of the Obama presidency arguing, unsuccessfully, for a much faster American drawdown in Afghanistan.

He is determined not to let the United States get sucked into direct combat with a nuclear-armed Russia, and spent the first two years of his presidency trying to pull back from the Middle East and focus more on the Indo-Pacific.

And so while American weaponry and intelligence are central to both wars, Mr. Biden is living with the reality that the military decisions must be made in Israel and Ukraine, not the United States. That often leaves Washington in an odd position, able to suggest techniques for collapsing the huge tunnel networks in Gaza or poking through Russian defensive lines, but distancing itself from the decisions and their aftermath.

“We’re not sitting down next to them as they develop the target lists,” John F. Kirby, a spokesman for the National Security Council, told reporters on Monday, when pressed on whether the United States, as the provider of many of Israel’s weapons, takes responsibility for the civilian casualties. “This is their fight.”

When not speaking on the record, some of Mr. Biden’s aides say the president has been taken aback by Mr. Netanyahu’s unwillingness to bend on the question of attacks on dense urban areas. When Secretary of State Antony J. Blinken arrived in Israel late last week and made the case again for humanitarian pauses — the best way to get aid into Gaza, civilians out of the line of fire and perhaps to facilitate the release of some prisoners — Mr. Netanyahu rejected the call out of hand.

At moments, they say, he and his war cabinet have responded to American military advice about how to conduct urban combat while minimizing civilian casualties by saying they are fighting with the weapons they have at hand — 1,000- and 2,000-pound bombs, some of the biggest in any military arsenal. Those weapons were never envisioned for use in a dense urban area, and the United States is trying to rush to Israel much smaller bombs, better suited for penetrating the tunnels without causing as much collateral damage.

But with each fresh batch of images of injured or killed children, the pressure on Mr. Biden is rising, with some members of his own party urging him to embrace a cease-fire — which is quite different, and more long-lasting, than episodic “humanitarian pauses.” Those calls are likely to accelerate after the Gaza health ministry estimated on Monday that Israeli strikes had killed about 10,000 Palestinian civilians, including roughly 4,000 children and teenagers. The ministry is controlled by Hamas, so the figures are impossible to confirm.

Jon Finer, the deputy national security adviser, contended in an interview on CBS’s “Face the Nation” on Sunday that the United States was using its influence, but quietly. “We build law-of-armed-conflict assurances anytime we transfer security assistance to any country, including Israel,” he said.

“And when we see circumstances, events, that concern us, we raise those very directly with the government of Israel. And again, we will continue to do that as this conflict transpires.”

The challenge of the Ukraine war is quite different, but equally complex. Here the pressure on Mr. Biden is not from the left; even some of the most progressive members of his party support sending tens of billions of dollars more in arms and other support to President Volodymyr Zelensky’s government, to enable it to fend off the Russian invasion 20 months ago.

But on the right, the support for that help is eroding quickly. And the administration is having a difficult time articulating what the next move is, after the long-promised “spring offensive” against Russia failed to move the battle lines more than a few miles.

General Zaluzhny said that it would take a major technological advance in weaponry to end the stalemate on the ground, and added that “there will most likely be no deep and beautiful breakthrough.” But it is unclear what that technology leap would look like.

Mr. Biden’s aides say they have now given Mr. Zelensky every weapons system he has requested, most recently ATACMS, the long-range missile systems Mr. Biden had long resisted providing because he feared they might cross a “red line” that could lead Mr. Putin to reach for nuclear weapons. Now the fear about the ATACMS is that they will not make that big a difference because the Russians are learning how to park their aircraft beyond the weapons’ reach.

Mr. Zelensky rebuked his own general for the “stalemate” characterization over the weekend, and complained anew that much of the American equipment arrived too late to make the kind of impact he needed. (Mr. Biden’s aides dispute that, saying they have provided weaponry when the Ukrainian forces could use them.) But Ukraine, American officials say, has ignored Pentagon advice about concentrating its forces to break through one or two strongholds in the Russian network of trenches and minefields, rather than spread them out thinly.

So now, Mr. Biden is trying to channel fatigue and frustration with the war in Ukraine, born of the growing sense that billions of dollars in American arms, aid and intelligence collection has simply failed to overcome the assembled weight of the dug-in Russian army.

“What I worry about,” Douglas Lute, a retired general who was central to putting together the Afghanistan strategy in the Bush and Obama administrations, said at an event at the U.S. Military Academy at West Point last week, “is that we are giving them enough to stay in the fight, but not enough to win.”

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