The Senate's top Democrat meanwhile is raising the possibility of "a lot of problems" if Mr. Bush settles on federal appeals judge Samuel Alito to succeed retiring Justice Sandra Day O'Connor, a swing voter on abortion and other social issues.
CBS News correspondent Joie Chen reports that whoever Mr. Bush picks has to satisfy both Conservatives and Democrats who might try to
Others said to be under consideration include federal appeals court judges J. Michael Luttig, Karen Williams, Priscilla Owen and Alice Batchelder as well as Michigan Supreme Court Justice Maura Corrigan.
A leading Republican senator suggested a staff shake-up at the White House, saying Harriet Miers' nomination might have failed partly because there "wasn't enough consultation or enough good, strong people ... advising the president."
"You've got to reach out and bring in more advice and counsel," said Republican Sen. Trent Lott.
A nominee perceived by Democrats as extremely right-wing could provoke a bitter confirmation fight, given the increasingly hardened positions over a woman's right to have an abortion, the Senate Judiciary Committee chairman said.
"You have both sides poles apart and insistent on finding some answer to that question in advance of the hearing, which no one is entitled to," said Republican Sen. Arlen Specter.
Specter told CBS News' John Roberts that he does not want the process to be decided by press conferences and
"I'm looking for an opportunity to ask questions of the nominee," Specter said. "To have a thorough processes, and then have the committee make the determination and have the senate vote."
With the expectation that Mr. Bush probably will turn to a sitting judge, it was Alito's name that was mentioned often on the Sunday talk shows.
A judge on the Philadelphia-based 3rd U.S. Circuit Court of Appeals, Alito has been dubbed "Scalito" or "Scalia-lite" by some lawyers because his judicial philosophy invites comparisons to Supreme Court Justice Antonin Scalia's.
"That is not one of the names that I've suggested to the president," Senate Minority Leader Harry Reid told "Late Edition" on CNN. "In fact, I've done the opposite. I think it would create a lot of problems."
Reid said Mr. Bush would be making a "mistake" were he to settle on a hard-liner simply to appease the far right in his party, especially after conservatives' wrath undermined Miers' nomination.
Reid and other Democrats said that Mr. Bush, given his low poll numbers, should nominate a consensus candidate rather than someone selected specifically to rally his conservative base.
"I think the American people can see through this so clearly. The president should come forward with some middle-of-the-road person, somebody that is going to be a good Supreme Court justice, not somebody that's going to be writing the law from the bench," Reid said on ABC's "This Week."