But unlike previous fights, the aide - Cheney Chief of Staff David Addington - has expressed a willingness to comply.
Conyers announced Thursday that his committee will convene next week to vote on whether to issue a subpoena. But the chairman issued his threat after the committee received a letter from Cheney's counsel, Kathryn Wheelbarger, suggesting Addington would consider the request.
The committee has asked Addington to appear at a hearing next Tuesday about the legal justifications for a series of controversial memos that laid the foundation for the administration's war against terror, including the detainment and treatment of suspected terrorists.
Conyers has also invited former Attorney General John Ashcroft and John C. Yoo, a former Justice Department lawyer in the office of legal counsel who drafted a number of the controversial briefs justifying the use of extreme interrogation methods. In one recently released memo from 2003, Yoo argued broadly that the president's war powers as commander-in-chief would trump other local or international laws.
Wheelbarger sent the committee a letter Thursday arguing that Addington should not be required to testify before Congress about his duties as an aide to the vice president. However, in the letter, she acknowledges that Addington "is prepared to accept timely service of a committee subpoena" for testimony at the May 6 hearing.
"Since he hasn't been issued a subpoena, it would be a little premature to comment on whether he would comply," said Cheney spokeswoman Megan Mitchell. But, in clarifying the intent of that letter, Mitchell said that if a subpoena is issued, Addington would "review it and respond accordingly."
The details of his appearance remain in flux, but a Conyers spokeswoman welcomed the development, saying the committee looks "forward to coming up with a mutually acceptable date for Mr. Addington's testimony."