The UK's current extradition deal with the United States should be scrapped until the US Senate signs its side of accord, the House of Lords has said.
Pleas for the men to be tried in the UK have been rejected
The government was defeated in the Lords as controversy continues over three British bankers due to be extradited to the US on Thursday.
Peers said the extradition arrangements should be put on hold - but their vote has no force without MPs' approval.
The vote came after MPs forced an emergency Commons debate on the issue.
In a highly unusual move, Commons Speaker Michael Martin allowed Wednesday's debate after a request by Liberal Democrat Nick Clegg.
A minister will have to be go to the Commons to put the government's case in the three-hour debate.
The former NatWest bankers - David Bermingham, Gary Mulgrew and Giles Darby - are due to be extradited on Thursday over allegations of fraud linked to collapsed energy giant Enron.
They deny any wrongdoing but have lost their court battles to face trial in the UK.
The US has not yet ratified the extradition treaty with the UK, which critics say means the arrangements are one-sided as the UK passed its laws in 2003.
But the government argues that the arrangement means both countries have to produce an equivalent level of legal evidence when seeking extradition from the other nation.
The 2003 Extradition Act means the US no longer has to put forward a prima facie case when asking for people to be extradited from the UK.
Peers demanded that the US be taken off the list of countries which benefit from the laws until the Senate ratifies the treaty.
Conservative former Attorney General Lord Mayhew said the vote would send a message "that the Brits are not patsies", said the former attorney general.
The Lords voted by 218 to 116 to suspend the deal.
But the change would not become law unless it were approved by the Commons, and the Police and Justice Bill is not due back there until October.
Liberal Democrat legal affairs spokesman Lord Goodhart argued the government ought to put the existing arrangements "on hold" until MPs had their say.
Later, the Lords inflicted a second government defeat in a change designed to make it easier to put people up for extradition on trial in the UK.
Home Office Minister Lady Scotland accused peers of trying to give the House of Commons a "slap" by pushing through the change.
She is flying to Washington on Wednesday to appeal for the Senate to ratify the extradition treaty, which was agreed back in 2003.
A Home Office spokeswoman said the government was disappointed by the votes.
"Were these amendments to become British law, a UK court would require less information to surrender a UK citizen to the legal system of Azerbaijan or Albania than the United States of America," she said.
"To do so would be to write into UK law a rebuke to the American legal system which has recently provided the checking power to the US President on Guantanamo."
Tony Blair has said he will try to secure bail for the bankers once they reach the US.
But as he urged the Commons debate, Mr Clegg said the prime minister's defence of the "lop sided" treaty posed more questions than it answered.
It was difficult to understand why ministers were only belatedly putting pressure on Washington to exercise its side of the bargain, he told MPs.