26 Labour MPs, including several prominent Yorkshire MPs, have signed a joint letter.
Commenting on the letter, Ms Flint said: “This is fundamentally about democracy. The country voted to leave the EU. They have a right to trust that MPs will implement the decision – a promise both Labour and the Conservative made at the general election less than a year after the referendum.
“I want to look my voters in the eyes and say ‘I tried my best to get Parliament to honour the decision to leave.’
“To hold a second referendum will cause huge anger and distrust of politicians, that they don’t keep their promises; their words mean nothing.
“Why should leave voters have to vote twice to see the decision honoured, but those voting remain only need to win once. Any decision by Labour to back a second referendum will drive a wedge between my party and many voters who have backed Labour their whole lives.
“All the MPs are asking is that we implement the decision to leave; and do so in an orderly way by supporting the best deal we can. Once we have done so, Labour can get back to the campaigning on many issues: policing, health, jobs, and poverty that do not get a hearing at the moment."