In a recent years, the chief crisis manager has been the President's national security adviser, but the choice of Mr. Bush was apparently dictated by the White House's desire not to give the position to Richard V. Allen, Mr. Reagan's national security adviser. Haig Role Is Not Outlined

No mention was made in the announcement of what role Mr. Haig might play. The personnel for Mr. Bush's committee would be supplied by Mr. Allen's staff, the announcement said.

Mr. Haig's remarks suggested the possibility that he might resign or take some other action if Mr. Bush was given the job. But a department spokesman was authorized to say tonight that Mr. Haig had no intention of resigning.

It was understood that President Reagan decided to go ahead with the choice of Mr. Bush to head the crisis-management team this afternoon after White House aides urged him not to let Mr. Haig's comments remain unchallenged. The President telephoned the decision to Mr. Haig late in the day.

The decision to place Mr. Bush in charge of crisis management followed reports that had circulated for several days that he would be given the same kind of responsibility given to Henry A. Kissinger and Zbigniew Brzezinski when they were national security advisers. Mr. Bush, a former director of Central Intelligence, also headed the American missions to China and the United Nations. Defense Dept. Approves Decision

The announcement of Mr. Bush's appointment was welcomed at the Defense Department. A Pentagon official said that Defense Secretary Caspar W. Weinberger was pleased with the choice and believed that Mr. Bush was ''thoroughly knowledgeable and experienced.''

Mr. Weinberger was said to believe that this would permit all information and ideas to get to the President without the bias of the State Department, Defense Department, Central Intelligence Agency or any other agency.