The former cabinet minister warned Brexit was too important to be a victim of the “chaos” in the Conservative Party following last week’s general election.
Article 50 of the Lisbon Treaty - the two-year process for leaving the European Union - was triggered before the election and talks with Brussels are due to begin within days.
The Prime Minister has insisted the Brexit talks will proceed as planned despite the Conservatives losing their majority.
But Theresa May faces questions over how the Government will be able to hold meaningful talks with EU leaders when it cannot guarantee support at home for any agreement reached.
Ms Cooper, last week re-elected MP for Normanton, Pontefract and Castleford for the sixth time, said: “I think the big challenge for us is actually now about the Brexit negotiations because there is not a lot we can do about the chaos in the Conservative Party at the moment.
“But Article 50 has been triggered and there needs to be some sort of brexit negotiation process take place.
“I’ve always said I thought that should be done as a cross-party process. I think there now should be a sort of cross-party commission or group set up to try and take forward those negotiations in a way that is open, that is thoughtful that is consensual, that accepts that not everyobdy is going to get the deal they want.
“There’s going to have to be compromises on all sides but if you try and do it in that way we might actually be able to pursue the national interest rather than it just getting stuck in this Tory small cabal, which is what Theresa [May] wants and is simply not going to work.”
Over the weekend, German Chancellor Angela Merkel made clear European leaders would enfoce the two year schedule.
“We are ready for the negotiations. We want to do it quickly, respecting the calendar We were waiting for the election in Britain, but in the next few days these talks will begin,” she said.