McCain, the Arizona senator whose previous efforts at an immigration overhaul ended in failure in 2007, predicted the White House proposal's demise if it were sent to Congress. He strongly urged the president to pocket the drafted measures.
"I believe we are making progress in a bipartisan basis," said McCain, who is in the Senate group working on legislation.
And Democratic Sen. Chuck Schumer, who met with Obama on Wednesday at the White House to discuss progress, urged his allies in the administration to give a bipartisan group of eight lawmakers the time to hammer out a deal on their own.
Schumer, a New York Democrat and a close ally of the White House, said he has not seen the draft proposals but, along with the Democrats working on a compromise, met with Obama this week to talk about progress being made on Capitol Hill.