Mr. Bush also selected Susan Schwab, the deputy trade representative, to move up to the top trade job, replacing Portman.
Tuesday's announcements are just the start of a shift in the White House senior staff lineup, reports CBS News correspondent Peter Maer.
"With a new man will come some changes," the president said.
That new man is chief of staff Joshua Bolten, the former budget director, who is already reshaping the White House team. Under pressure from members of his own party who want a fresh approach from the administration, the president says more changes are in the works.
But, Mr. Bush said, those changes will not include a replacement for Defense Secretary Donald Rumsfeld, despite calls for his resignation from a half dozen retired military commanders.
"I hear the voices and I read the front page and I know the speculation," the president told reporters at a Rose Garden ceremony. "But I'm the decider and I decide what's best. And what's best is for Don Rumsfeld to remain as the secretary of defense."
At a Pentagon news conference later in the day,and added: "The president knows, as I know, there are no indispensable men ... He knows that I serve at his pleasure, and that's that."
The president noted that Washington is buzzing with rumors about an administration shake-up. Treasury Secretary John Snow is said to be on the verge of leaving, and Republicans outside the White House say they expect changes in the White House lobbying and communications shops.
"I understand this is a matter of high speculation here in Washington," the president said. "It's the game of musical chairs, I guess you would say, that people love to follow."
He said he had given Bolten broad authority to make changes to strengthen the White House.
"And of course he will bring different recommendations to me as to who should be here and who should not be here," the president said.
Mr. Bush said that while Washington is fixated on gossip about personnel changes, the administration is dealing with problems such as soaring gasoline prices and the war on terror.
"I'm concerned about higher gasoline prices," the president said. He said the government has responsibility "to make sure that we watch very carefully and investigate possible price-gouging."