Gay and lesbian advocates were also expecting Obama's proposals to include recognition of same-sex couples where one partner is American and another is not.
Sen. John McCain called the issue a "red flag" in an interview Tuesday on "CBS This Morning."
The Arizona Republican also said he didn't think the issue was of "paramount importance at this time."
"We'll have to look at it," McCain said. But he added that the highest priority is finding a "broad consensus" behind the immigration bill already being planned. He said the country must do something about 11 million people "living in the shadows."
Obama's previous proposals for creating a pathway to citizenship required those already in the U.S. illegally to register with the government and submit to security checks; pay registration fees, a series of fines and back taxes; and learn English. After eight years, individuals would be allowed to become legal permanent residents and could eventually become citizens five years later.
The Senate group's pathway to citizenship for illegal immigrants already in the U.S. would be contingent upon securing the border and improving tracking of people in the U.S. on visas. Linking citizenship to border security could become a sticking point between the White House and lawmakers.
The Senate framework would also require those here illegally to pass background checks and pay fines and taxes in order to qualify for a "probationary legal status" that would allow them to live and work here - but not qualify for federal benefits - before being able to apply for permanent residency, a critical step toward citizenship. Once they are allowed to apply they would do so behind everyone else already waiting for a green card within the current immigration system.
Passage of legislation by the full Democratic-controlled Senate is far from assured, but the tallest hurdle could come in the House, which is dominated by conservative Republicans who've shown little interest in immigration reform.
The senators involved in formulating the immigration proposals, in addition to McCain, are Democrats Charles Schumer of New York, Dick Durbin of Illinois, Robert Menendez of New Jersey and Michael Bennet of Colorado; and Republicans Lindsey Graham of South Carolina, Marco Rubio of Florida and Jeff Flake of Arizona.
Several of these lawmakers have worked for years on the issue. McCain collaborated with the late Democratic Sen. Edward M. Kennedy on comprehensive immigration legislation pushed by then-President George W. Bush in 2007, only to see it collapse in the Senate when it couldn't get enough GOP support.