Israeli forces moved to seize the entirety of a key roadway bisecting the Gaza Strip on Wednesday, the Israeli defense minister said, effectively dividing the enclave’s north from its south as thousands of Palestinians continued to flee Israel’s offensive in Gaza City.

The Israeli advance came as Hamas held talks in Qatar with mediators over President Trump’s plan to end the war in Gaza. Mr. Trump announced the proposal at a news conference on Monday alongside Prime Minister Benjamin Netanyahu of Israel, who said he backed it.

If the Palestinian militants do not accept the deal, Mr. Trump said, Israel will have “my full backing to finish the job of destroying the threat of Hamas.” And on the ground in Gaza, the fighting has continued.

On Wednesday, Israel Katz, the Israeli defense minister, said Israeli troops were “currently completing the takeover” of the central Netzarim Corridor, the key roadway. He added that doing so would “cut Gaza in two between north and south.”

Hundreds of thousands of Gazans have fled the Israeli assault on Gaza City and crowded into swelling tent camps farther south. But hundreds of thousands more are believed to remain in the northern city, with many saying that they have nowhere left to run to or cannot pay for transport out. Others had fled south only to return to Gaza City after failing to find anywhere suitable to take shelter.