Even as President Biden presses Israel to define clearly the goals of its war against Hamas in Gaza, he is turning his eyes to a much larger endgame: the ever-elusive hope for a lasting peace deal between Israel and the Palestinians.

Speaking to reporters last week, Mr. Biden said that “when this crisis is over, there has to be a vision of what comes next, and in our view it has to be a two-state solution,” creating a sovereign Palestinian nation alongside the state of Israel.

The question is how hard Mr. Biden intends to work for that outcome.

Until last month, the Israeli-Palestinian conflict was not among his top priorities. A president focused on countering China and then Russia’s invasion of Ukraine had little time or inclination for a distant goal that stymied — and politically bruised — several of his predecessors.

Biden administration officials also doubted whether Israel’s increasingly hawkish leadership was interested in any plausible deal. They also wondered whether the Palestinians would trust the United States as a peace broker after four years of the Trump administration’s dramatically pro-Israel tilt and a Trump peace plan that the Palestinian Authority leader Mahmoud Abbas, excluded from its devising, declared dead on arrival.

Unlike his recent predecessors, Mr. Biden did not appoint a special envoy for Middle East peace or task his secretary of state with trying to forge an agreement between Israel and the Palestinians. Instead, he focused on mediating a normalization agreement between Israel and Saudi Arabia, hoping to leverage Israeli concessions to the Palestinians along the way.

“It was not a tier-one policy objective to solve this conflict,” said David Makovsky, a former peace process negotiator in the Obama administration. “The goal was to stabilize, and not trying and failing a fifth time.”