Mr Bush paid tribute to Justice Rehnquist, who was the most senior judge of the US Supreme Court, describing him as a man of “character and dedication.”
He praised him for his “powerful intellect” and “profound sense of duty.”
The president said his death is a great loss for the court and the country.
Appointed to the court by former US President Richard Nixon in 1972, his death will spark an intense political battle over his replacement.
Mr Bush must now fill two vacancies on the Supreme Court following the retirement of another top judge several months ago.
He said he plans to move fast to find new judges for the court, saying it is in the best interest of the nation to fill the vacancies promptly.
The replacement of two judges could fundamentally change the court’s ideological balance and provoke a significant political battle.
Mr Bush has already named conservative federal appeals court judge John Roberts to replace Sandra Day O’Connor, who is retiring.
Democratic senators are promising to challenge Mr Roberts on key issues like abortion rights, privacy and the role of government.
They are expected to scrutinise his credentials as they decide whether to approve of his nomination.
Mr Bush could choose to elevate one of the current justices to the chief justice position, such Antonin Scalia.
Mr Scalia is an ultra-conservative recognised as one of the country’s most brilliant jurists.
“Scalia is the favorite of hard core conservatives,” said law professor Jonathan Turley, a regular critic of the Bush administration.
Another frontrunner is moderate conservative Attorney General Alberto Gonzales.
He is thought to have been a candidate to replace Sandra Day O’Connor.
But legal analysts believe Mr Bush may want to steer clear of controversy amid criticism of America’s handling of Hurricane Katrina.
Other observers speculate that Mr Bush might decide to appoint Mr Roberts directly to the chief justice post.
In any case, liberals fear that the president will use the two vacancies to fashion one of the most conservative courts ever in American history.