President Biden and Prime Minister Benjamin Netanyahu of Israel each addressed the future of the battle in Gaza this week, speaking just a day apart but worlds removed from one another in a way that captured the essential tension between the two men after more than four months of fighting.

Mr. Netanyahu spoke of war and how it would continue even if there is a temporary cease-fire to secure the release of hostages, just “delayed somewhat.” Mr. Biden spoke of peace and how such a cease-fire deal could “change the dynamic,” leading to a broader realignment that would finally end the underlying conflict that has defined the Middle East for generations.

The disparity in visions reflects the opposing political calendars on which the two leaders are operating. Mr. Netanyahu has a compelling interest in prolonging the war against Hamas to postpone the day of reckoning when he will face accountability for failing to prevent the Oct. 7 terrorist attack. Mr. Biden conversely has a powerful incentive to end the war as soon as possible to tamp down anger in the left wing of his party before the fall re-election campaign when he will need all the support he can get.

At the same time, each has reason to think he may yet get a better deal if the other loses his post. Mr. Biden’s advisers are acutely aware that Mr. Netanyahu’s government could fall in response to the terrorist attack while the Israeli prime minister, who goes by the nickname Bibi, may prefer to buy time until November in case former President Donald J. Trump recaptures the White House.

“It’s absolutely fair to say Biden and Bibi are on different political timetables with respect to the Gaza war — and I think it’s an increasingly significant part of the equation,” said Frank Lowenstein, a former special envoy for Middle East peace under President Barack Obama.