President Trump’s latest foray into Middle East peacemaking presented Hamas with what sounded like an ultimatum. He pressed the Palestinian militant group to either accept a new American cease-fire proposal or face the full wrath of Israel’s military advance into Gaza City.

“The Israelis have accepted my Terms. It is time for Hamas to accept as well,” Mr. Trump posted on social media on Sunday, hinting at a new American proposal to exchange all the remaining hostages for Palestinian prisoners and end the nearly two-year-old Gaza war.

“I have warned Hamas about the consequences of not accepting. This is my last warning, there will not be another one!” Mr. Trump said.

It was not immediately clear whether Mr. Trump’s brinkmanship could stop the invasion of Gaza City, the main urban center of the Palestinian territory. Alternatively, if Hamas balks, it could allow Israel and the United States to argue that they had tried everything and the group was bringing disaster upon itself.

The intervention did succeed in adding to the uncertainties surrounding Israel’s impending ground assault on the heart of Gaza City in the north of the territory. Hundreds of thousands of Palestinian residents, many of them already displaced at least once by the war, are now torn between fleeing to the overcrowded south or taking the risk of staying put.