Thompson, 62, a Republican and actor on the NBC television series "Law & Order," agreed to accept the post in a telephone conversation with the president on Monday, spokesman Scott McClellan said.
He said Thompson would serve as an informal adviser to shepherd the nomination through the Senate.
Ed Gillespie, former Republican National Committee chairman, also will help the administration push the nominee. He is expected to help with strategy, lobbying and news media relations.
The announcement came as Mr. Bush, traveling in Denmark, urged senators to act "in a dignified way" in what is expected to be a contentious battle over confirming his first nominee to the nation's highest court.
Mr. Bush was visiting Denmark to thank the Danes for sending troops to Iraq and Afghanistan. He also strongly defended his decisions on Iraq, climate change, imprisoned terrorism suspects and aid to Africa — all of which have made him unpopular in Europe.
"I understand that people aren't going to agree with decisions I make," Mr. Bush said as he stood alongside Danish Prime Minister Anders Fogh Rasmussen outside a white mansion that serves as his official summer residence. "I truly believe we're laying the foundation for peace."
Mr. Bush said that as he reviews candidates to replace, "I'll try to assess their character, their interests."
Mr. Bush said he would have no "litmus test" that disqualifies candidates because of their opinions on abortion and gay marriage.
"I'll pick people who, one, can do the job, and people who are honest, people who are bright and people who will strictly interpret the Constitution and not use the bench to legislate from," he said.
He has said that he will spend a few weeks narrowing a list of candidates and then interviewing some, and his goal is to see a new justice in place by the time the court begins its new term in October.
"I will take my time," Mr. Bush said. "I will be thorough in my investigation."