Bush on Monday called on the Senate to confirm him before the top US court opens its term on 3 October.
Roberts, who once clerked for Rehnquist, said he would be honoured "to succeed a man I deeply respect and admire".
The swift move would promote to the Supreme Court's top job a man who is being considered as one of eight associate justices.
"I am honoured and humbled by the confidence the president has shown in me," Roberts said, standing alongside Bush in the Oval Office.
"He's a man of integrity and fairness and throughout his life he's inspired the respect and loyalty of others," Bush said.
"John Roberts built a record of excellence and achievement and reputation for goodwill and decency toward others in his extraordinary career."
The selection of Roberts, who has drawn little criticism, helps Bush avoid new political problems when he already is under fire for the government's sluggish response to Hurricane Katrina and the president's approval ratings are sagging.
Bush is under fire for his handling
of the Hurricane Katrina disaster
Getting a new chief justice of Bush's choosing in place quickly also avoids the scenario of having liberal Justice John Paul Stevens making the decisions about whom to assign cases to and making other decisions that could influence court deliberations.
As the court's senior justice, Stevens would take over Rehnquist's administrative duties until a new chief is confirmed.