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...