When Prime Minister Benjamin Netanyahu of Israel visited the White House two months ago, President Trump sold him a brazen dream: The United States would take control of the Gaza Strip, relocate about two million Palestinians and turn the devastated seaside enclave into a glittering “riviera.”

This week, as the two leaders faced reporters again after meeting in the Oval Office, Mr. Trump appeared to have moved on, holding forth instead on U.S. border policy, his new tariffs, the plight of the hostages held in Gaza and the latest showstopper for Middle East policy — the opening of talks with Iran to curb its nuclear weapons program.

But Mr. Netanyahu did not let the Gaza idea — however unfeasible or potentially illegal — fade like a mirage. He raised it himself, saying that he and Mr. Trump had discussed the vision, including which countries might agree to accept Gazans.

Mr. Netanyahu and his government say they are serious about the idea but emphasize that they are speaking about facilitating the “voluntary” migration of Palestinians, in an apparent attempt to avoid any suggestion of ethnic cleansing. Critics say that it would hardly be voluntary if Gazans left, regardless, given that so many of their homes have been smashed to rubble.

Days after Mr. Trump’s original announcement, the Israeli defense minister, Israel Katz, said he was establishing a special administration within the ministry focused on voluntary migration from Gaza. In late March, he appointed a senior ministry official, Yaakov Blitshtein, to head it.