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. …

Mr. Rubio, saying immigration would be a top priority for him this year, said he was currently laying out principles and gathering support for them. As a tactical matter, Mr. Rubio is not delving too deeply into the details at this stage.

His plan would give a temporary “nonimmigrant visa” to illegal immigrants, which would allow them to remain and work in the United States. They would have to wait a “significant but reasonable” period of time before they could apply to become legal permanent residents, going to the back of the line in the existing system. Once they became residents, they could go on like other legal immigrants to naturalize as citizens.