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."
Bolten, at his first senior staff meeting on Monday, pointed toward more personnel changes. He said that staffers who were thinking about leaving before the end of the year should go now.
A former Republican congressman from Ohio, Portman is a confidant of Mr. Bush's and has broad experience in Washington.
Mr. Bush said Portman will be a "powerful voice for pro-growth policies and spending restraint."
Accepting the nomination, Portman said, "It's a big job. The Office of Management and Budget touches every spending and policy decision in the federal government."
As a member of the tax-writing House Ways and Means Committee and as vice chairman of the Budget Committee, Portman forged bipartisan coalitions on legislation dealing with tax, trade and pension reform policy.
It was just a year ago that Mr. Bush tapped Portman, a Republican congressman from Ohio, to succeed Robert Zoellick as his top trade negotiator.
Portman had served in the House since winning a special election in 1993. He was highly regarded for his skills in forging compromises between Republicans and Democrats, traits that he will be able to use in his new job as budget director.
Portman won praise during his brief time in the trade job for his role in winning passage of the Central American Free Trade Agreement with six Latin American nations.
The pact was vigorously opposed by Democrats, who argued that it would open American workers to more unfair competition from low-wage nations. Portman, however, overcame that opposition by convincing enough wavering Republicans to support the agreement.
Portman pledged during his confirmation hearings last year to take a tougher line with China and in February his office produced a top-to-bottom review of U.S. relations with China which the administration is expected to use to push for more trade concessions when Mr. Bush meets with Chinese President Hu Jintao on Thursday at the White House.