That position gives "us no choice but to proceed with a subpoena," said Henry Waxman, the House oversight committee's democratic chairman.
The administration says that senior officials, including Rice, have answered questions about the now-discredited claim that Saddam Hussein, Iraq's former president, had been seeking uranium for a bomb.
The subpoena adds to a series of investigations which include the dismissals of eight federal prosecutors by Alberto Gonzales, the US attorney general.
"I am beginning to wonder whether the White House has any interest in the American people learning the truth about these matters"
Patrick Leahy, senate judiciary committee chairman
The House judiciary committee voted 32-6 to grant immunity from prosecution to Monica Goodling, Gonzales' White House liaison, for testimony about why the administration fired the prosecutors.
The panel also unanimously approved - but did not issue - a subpoena to compel her to testify.
She has said previously she would not testify because she fears incriminating herself, which the US constitution insures Americans against.
The committee scheduled a May 10 hearing for Gonzales.
Democrats say the round of investigations represent a revival of congress' role after six years during which Republican majorities exercised control.
The White House has responded by refusing to allow officials to testify under oath about the firings of the prosecutors, despite criticism of the dismissals by many Republicans.
"I am beginning to wonder whether the White House has any interest in the American people learning the truth about these matters," said Patrick Leahy, senate judiciary committee chairman and Democrat.
Arlen Specter, ranking Republican on the judiciary committee, co-signed a letter with Leahy on Wednesday urging Gonzales to freshen his memory over the sackings and provide answers within a week.