The move would give President Bush the rare opportunity to fill two seats on the nation's highest court and likely set off a brutal Senate confirmation battle for whomever he nominates for the openings.
The 80-year-old Rehnquist has cancer and his retirement has been widely anticipated.
On Friday, he was asked directly by reporters if he was about to resign.
"That's for me to know and you to find out," replied the chief justice.
He didn't retire on Friday, which set off rumors that he would do it this week, perhaps as soon as Monday.
Despite the speculation, the Republican chairman of the Senate Judiciary Committee told Bob Schieffer Sunday on CBS News' Face the Nation that he doesn't think Rehnquist will step down.
"I doubt it very much, Bob," said Sen. Arlen Specter, R-Pa.
"My own analysis is that the chief does not intend to step down as long as his health holds out. Having been engaged in a bout with cancer myself, I know that it's good to get up every morning and have something that you have to do, something that is important to do."
Specter said that if Rehnquist's health holds up, he'll probably stay on the job. But, he added, "I believe he may not know that, really, from one day to the next, one week to the next, one month to the next. So there's not much he can say to the press."
Sen. Pat Leahy of Vermont, the top Democrat on the Judiciary Committee, agreed.
"I think the chief justice wants to stay," said Leahy. "I wish he could be up at his home right now in Greensboro, Vermont. He loves the place. It's a gorgeous area. I think that would do wonders for him. I suspect his doctors say stay closer to Washington."
With one vacancy on the high court already, the senators are going to the White House on Tuesday for discussions with President Bush.
"I hope it's real consultation and not just check off the box, have some of the leaders down for breakfast," said Leahy. "I commend the president for picking up the phone - he called Arlen, he called myself, within an hour of time that Justice O'Connor resigned."
Leahy said consulting with senators could help the president. "Obviously, he's the one that makes the choice. He's the one who makes the nomination, not any group of senators. But it could help him very much in having a nominee who could unite the country, not divide the country.
"After all, the Supreme Court's there for all of us. And I would hope that he would really actively seek our advice on somebody who might unite us."
Asked if they would talk about specific nominees with the president, Leahy said, "I think if he wants that kind of input, I think he will get it. But he will get it privately. I said to somebody earlier this morning, I probably won't quote a name on this program or any other program, because I don't want that to be the kiss of death for the nominee."
Specter was asked if having two vacancies to fill would change the president's selection process.
"I think it would change the equation." Specter said.
"One line of speculation is that it would give the president a chance to put somebody whom the conservatives would really like very much to fill where Rehnquist has been philosophically on the court, and somebody who is more of a swing voter, like Justice O'Connor."
Specter also said there had been some speculation that O'Connor might be willing to remain on the court if she has a chance to replace Rehnquist as chief justice.
"There had been a fair amount of talk about that possibility. And in her letter of resignation, she conditioned it on the confirmation of a successor. So there's some flexibility. Who knows? Some speculation is that she might reconsider if she were named chief justice," Specter said.
"I think it would be very tempting if the president said, 'Justice O`Connor, you could help the country now.' She has received so much adulation that a confirmation proceeding would be more like a coronation, and she might be willing to stay on for a year or so."