WASHINGTON—President Donald Trump proposed a path to citizenship for 1.8 million undocumented immigrants brought to the U.S. as children, if lawmakers agree to create a $25 billion fund to expand barriers along the Mexican border and implement other deep changes to the immigration system.

The White House suggested the proposal moved Mr. Trump a step closer to the Democrats, who have championed the cause of the young immigrants known as Dreamers, but the plan includes demands they have fiercely opposed. The Trump plan also risks the ire of hard-liners who oppose any such pathway to citizenship.

The White House proposal, presented to Senate leaders and congressional aides on Thursday, would restrict family-based immigration, the channel by which most immigrants have come to the U.S. for the past half-century. It also calls for an end to a lottery program that randomly awards 50,000 green cards annually to foreigners from countries underrepresented in U.S. immigration.

Mr. Trump was in Davos, Switzerland, for the World Economic Forum on Thursday. His top aides told Senate Majority Leader Mitch McConnell that the president would sign into law legislation that included these changes. Mr. McConnell said he would bring a bill to a vote during the week of Feb. 5, White House officials said.

It wasn’t clear whether Mr. McConnell could find sufficient support. To reach the 60 required votes for such measures to pass, he will need backing from most Republicans, who hold 51 seats, and some Democrats. Passage in the House was also in doubt, with opposition from Democrats expected over policy changes and border-wall funding, and opposition from conservatives over a pathway to citizenship for undocumented immigrants.