Since Mr. Bush's first campaign for president, court watchers have been talking about how he could have the opportunity to shape the aging court. But he has been tightlipped when asked for specifics about whom he would pick.
"I'm not telling you," he told a questioner who asked for names in a debate with rival John Kerry last year. "I really haven't picked anybody yet."
Mr. Bush has said he admires Justices Antonin Scalia and Clarence Thomas, the two most conservative members of today's generally conservative court. Both would overturn the Roe v. Wade decision that legalized abortion, drop racial affirmative action and allow almost any government aid to religious schools.
Mr. Bush has a record of putting forward similarly conservative judges for lower courts. Justice Sandra Day O'Connor's announcement last week that she would retire gives Mr. Bush his first chance to nominate a judge to the highest court.
Sen. Orrin Hatch, R-Utah, a member of the Senate Judiciary Committee that will hold hearings on the nomination, said Tuesday the Senate was probably "in for a pretty partisan battle," particularly if the president nominated a "strong conservative" for the post.
"The president is going to choose a conservative," Hatch said on ABC's "Good Morning America," but added, "I don't think he's going to choose a right-wing conservative."
Democrats have indicated that a hard-line conservative would trigger a furious battle on Capitol Hill that could touch off a filibuster against the nomination.
Mr. Bush usually talks in general terms about what he will look for in a Supreme Court nominee, saying he wants someone who will "strictly interpret the Constitution" and "not use the bench to write social policy."
Clearly, there are some specific stances that Mr. Bush will examine. For example, he has said he will not choose someone who would say the Pledge of Allegiance should be banned in public schools because it contains the words "under God."
There are other issues important to Mr. Bush's conservative Christian base that he has signaled he will consider. In his acceptance speech at the Republican national convention last year, he criticized judges he contends have gone too far in rulings declaring gay marriage legal.
"I support the protection of marriage against activist judges," the president said, "and I will continue to appoint federal judges who know the difference between personal opinion and the strict interpretation of the law."