Benjamin Netanyahu made the announcement as he attended a dinner at the White House, and the US president appeared pleased by the gesture.
“He’s forging peace as we speak, and one country and one region after the other,” Mr Netanyahu said as he presented the US leader with a nominating letter.
Mr Trump took credit for brokering a ceasefire last month in Iran and Israel’s so-called “12-day war” – and so far the truce appears to have held.
Mr Netanyahu said the US and Israel were working with other countries who would give Palestinians “a better future” – and indicated those in Gaza could move elsewhere.
“If people want to stay, they can stay, but if they want to leave, they should be able to leave,” he added.
Indirect negotiations with Hamas are currently taking place in the hope of securing a 60-day ceasefire in Gaza, which could lead to the release of Israeli hostages in the territory.
Mr Trump has suggested an agreement could be finalised this week.
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