Justice Sandra Day O'Connor announced her retirement this week. The top U.S. court has nine justices, who hold their posts for life. The president chooses a judge to fill a vacancy, but the nominee must be approved by the Senate.
One name with some early momentum is current Attorney General Alberto Gonzales, a longtime friend of President Bush. But some conservative groups are already aligning against a possible Gonzales nomination, CBS News Correspondent Tony Guida reports.
Gonzales is considered too moderate by right-wing Republicans.
"I don't think the social conservatives ought to prejudge Attorney General Gonzales. Attorney General Gonzales may not even be in the picture," Senate Judiciary Committee Chairman Arlen Specter, a Republican, said on ABC television.
Specter said he thinks Bush will consider "the gender factor" in making a selection, and the fact that retiring O'Connor was a pivotal swing vote.
However, it is unclear what the president will consider.
Bush has said in the past that conservative Justices Antonin Scalia and Clarence Thomas are the kind of people he admires on the Supreme Court.
Sen. Orrin Hatch, R-Utah, former chairman of the Senate Judiciary Committee and himself a noted conservative, left no doubt about how he views Gonzales.
"He's a very fine man and if he gets picked I am certainly going to support him," Hatch told CBS News' Face The Nation.
Bush was at the presidential retreat, Camp David, mulling over his options, while Democratic Sen. Charles Schumer was urging a summit between senators from both parties and the president.
Asked whether he would support a filibuster — a protracted debate intended to stall action — if a hard-liner is chosen, the judiciary committee's ranking Democrat, Sen. Patrick Leahy, said, "I would hope that we don't reach that point."
"That's why we're going to meet with the president in about a week, going to urge that he put somebody who would unite the country, not divide the country," said Leahy, who was appearing on NBC television.
"I have no intention of filibustering, but it depends on who the president sends," said Democratic Sen. Joseph Biden told Face The Nation.
On ABC, Democratic Sen. Edward Kennedy jumped in to answer a question intended for Specter, prompting Specter to joke, "I heard the filibuster starting a little early."
With Republicans holding power in the White House and Congress, conservatives see the Supreme Court as the final obstacle to control of all branches of the federal government.
Liberals say that given O'Connor's swing position on the court, Bush must choose a consensus conservative — a move that would risk alienating the president's far-right base but would avoid a political war.
Still a powerful memory for the Senate is the 18-year-old fight that ended in defeat for Republican Supreme Court nominee Robert Bork, who was seen as too extreme in his conservative views.
"I wish we could drop these political terms for the court because the court is all too political now," Bork said on CNN.
Meanwhile, Gonzales, the current attorney general, deflected questions about his interest in the Supreme Court, but said he wasn't bothered by the criticism.
"Many of the people speaking probably don't have all the information about prospective nominees. What's important is what the president of the United States thinks about me," he said while en route to Iraq, where he made a surprise visit to American troops Sunday. "That's evident by the position he has asked me to fill."
Gonzales will help screen nominees even though he is widely thought to be under consideration by Bush to replace O'Connor. When asked by reporters if he was flattered to be considered a possible nominee. "I just look at the job that I do as attorney general. I'm happy in that job," he said.
Gonzales, 49, the son of Mexican migrant farm workers, would be the high court's first Hispanic justice.