In a meeting in New York with reporters and editors of The New York Times, Mr. Rubio said that any broad immigration legislation should create a nationwide exit system to check foreigners out of the country, to confirm that they left before their visas expired. He noted that at least 40 percent of an estimated 11 million illegal immigrants in the country came on legal visas but then overstayed.

Mr. Rubio said he would also insist on a nationwide program for employers to verify the legal authorization of new workers, although he did not specify whether he would favor an expansion of an existing federal electronic worker verification program or seek to create a new one…

He acknowledged that major pieces of his plan remain to be worked out. According to current federal visa rosters, most Mexican-born immigrants applying to become permanent residents now face a wait of at least 17 years to receive their document — known as a green card — even if they followed the rules and were approved. Mr. Rubio’s proposal could add seven million more Mexican immigrants to those backlogs. The path to citizenship he proposes for illegal immigrants could be several decades long.

“I don’t have a solution for that question right now,” Mr. Rubio said.