Republican Sen. Rand Paul on Tuesday called for an overhaul of immigration laws that would allow millions of undocumented immigrants to remain permanently in the U.S., a move that could add momentum to immigration legislation but risks angering his grass-roots supporters.

While Mr. Paul was careful not to say the word "citizenship" in his speech Tuesday, his plan would allow immigrants to gain legal status that could eventually put them on a path to apply for U.S. citizenship. [...]

In a speech peppered with Spanish phrases and anecdotes about his youth in Texas, Mr. Paul dismissed some conservatives' concerns about illegal immigrants.

"Growing up in Texas, I never met a Latino who wasn't working," Mr. Paul said. "I've never met an immigrant looking for a free lunch."