President Trump’s statements on Tuesday about an American takeover of the Gaza Strip and displacing millions of Palestinians were immediately dismissed by many as reckless and half-baked pronouncements, a provocative threat that Mr. Trump was unlikely to enforce.

At the same time, his comments are the latest example of how government officials on the right in both the United States and Israel now speak publicly about a shared goal: the takeover of Palestinian land.

The question of whether the West Bank and the Gaza Strip — territories captured and occupied by Israel in 1967 — might become the foundation of a future Palestinian state has been at the center of decades of failed diplomacy, bedeviling American presidents, Palestinian leaders and Israeli prime ministers.

While the prospects for this future dimmed long ago, Mr. Trump’s election has newly emboldened right-wing ministers in Prime Minister Benjamin Netanyahu’s government, and some of Mr. Trump’s own appointees, to speak publicly about Israel’s right to fully take over the West Bank.

“It’s the most right-wing government that we’ve ever had in Israel — and there never was a U.S. administration that shared these views to this extent, either,” said Itamar Rabinovich, a former Israeli ambassador to Washington.

Days after Mr. Trump’s election, Bezalel Smotrich, Israel’s finance minister, whom Mr. Netanyahu has given broad authority over the West Bank, said Mr. Trump’s return to the White House meant that “the year 2025 will, with God’s help, be the year of sovereignty in Judea and Samaria,” using the biblical name for the territory that makes up the West Bank.