But if he pushes too hard to the right, he's likely to touch off a titanic battle with Senate Democrats who have vowed to block any nominee they view as straying too far from the mainstream.
If the president does decide to avoid an all-out fight with the Democrats, Attorney General Alberto Gonzales would be a logical choice. Mr. Bush is known to think highly of Gonazles, who has a more moderate image than many others whose names have been mentioned.
Gonzales' elevation would also give the GOP a political boost with Hispanics, whose political strength and voting power has been making steady strides.
Weekly Standard editor William Kristol, who last week correctly predicted that O'Connor -- and not Chief Justice William Rehnquist -- would be leaving the court -- thinks that Mr. Bush will go with Gonzales.
"President Bush will appoint Attorney General Alberto Gonzales to replace O'Connor," Kristol wrote. "Bush certainly wants to put Gonzales on the Supreme Court. Presidents usually find a way to do what they want to do."
Kristol, a conservative, expects Mr. Bush to placate fellow conservatives with two main arguments:
Whether conservatives would buy into these kind of arguments remains to be seen.
Others mentioned in speculation are U.S. Appeals Court Judges J. Michael Luttig and J. Harvie Wilkinson III, both of the 4th Circuit Appeals Court in Richmond, Va. Also, judges John Roberts of the U.S. Court of Appeals for the District of Columbia Circuit, Michael McConnell of the 10th Circuit and
Samuel Alito of the 3rd Circuit.
Also: former Solicitor General Theodore Olson, former Deputy Attorney General Larry Thompson, Judge Edith Jones of the 5th Circuit and Judge Danny Boggs of the 6th Circuit.
Political observers are split on the question of whether Mr. Bush can name a conservative to the court without touching off a major political firestorm.
Passions are running so high that "there is going to be a presumption against any presidential nominee on the part of the Democrats, which is too bad," said Charles Black, a longtime GOP consultant who is close to the Bush family. "I think he could nominate Ted Kennedy, and they'd still go into full battle mode against him."
But Democratic consultant Joe Lockhart, former press secretary to President Clinton, said there doesn't have to be a knockdown, drag-out fight if Mr. Bush will confer with Senate leaders of both parties before submitting his choice - a practice Lockhart said was utilized by both presidents Reagan and Clinton for choices who were confirmed.
"If President Bush uses the model created by Reagan and Clinton, there is no inevitability of a big fight. But if he chooses to go it alone, it increases the likelihood that this will get caught up in partisan back-and-forth," Lockhart said.