"I trust Dick Cheney's judgment; I know his character," Bush said. "It's obviously an important decision for me and I can't think of a better person to work with on making that decision."

Bush, the presumptive Republican presidential nominee, said he hasn't made any list of potential prospects and Cheney's service as chairman would not disqualify him from being selected himself.

"No lists yet, but it is the beginning of the list," the Texas governor said.

Bush said the main qualification for his running mate would be "somebody who can be president of the United States." In the past, he has also said he wants his vice president to share his government philosophy.

Asked whether he might be leaning toward a woman or perhaps a Washington outsider, the governor replied: "They're all equally important as I stand here today."

Cheney suggested he would cast a wide net.

"Fortunately, there are many good candidates to chose from in our party," Cheney said in a statement. "We will look at them all. And we will make sure we have the best ticket possible this fall."

On Wednesday, Bush was spending the day in Washington. The governor, whose foreign policy credentials have been questioned by critics, is scheduled to meet with Russian Foreign Minister Igor Ivanov, in town to see President Clinton and other U.S. officials. The governor also was raising a record $18 million for the Republican Party at a dinner.

Bringing in Cheney to steer the search will give the campaign a dose of Washington and national political experience. After Bush lost the New Hampshire and Michigan primaries, some Republican officials questioned whether his Texas-based campaign team had the experience necessary for a general election campaign.

While Bush "doesn't want to make the 2000 election a referendum on his dad, you need graybeards, people who have a cachet of experience and savvy," said Stuart Rothenberg, a Washington political analyst.

Cheney, 59, is chief executive officer of Halliburton Co., a Dallas-based engineering and construction company.

He attended Bush's alma mater, Yale University, for two years before transferring to the University of Wyoming. In 1974, he became President Ford's chief of staff. In 1978, he was elected to Congress from Wyoming. He was re-elected five times and rose to the rank of minority whip -- the No. 2 Republican official.

In 1989, President Bush selected Cheney to be his defense secretary, a position he held during the Persian Gulf War.

The search for a running mate for Bush's Democratic rival, Vice President Al Gore, is headed by Warren Christopher, who helped President Clinton choose his first Cabinet and then served as secretary of state.

Speculation on Bush's running mate has focused on a number of Republican governors and his former GOP rival John McCain. Other possibilities include another former rival, Elizabeth Dole, and Colin Powell, who was chairman of the Joint Chiefs of Staff in his father's administration.

Will the former president have a role in the decision?

"He will not be a part of the formal process, but he does not need to be," Bush said. "He's my dad, I talk to him all the time. He's got a few suggestions to make on occasion on different subjects, and sometimes I listen to him and sometimes I don't."