Friends of David Burrowes, Gary McKinnon's local MP and an aide to Owen Paterson, the Environment Secretary, said he would be forced to quit if Mr McKinnon was sent for trial in the US.
Theresa May, the Home Secretary, will announce whether Mr McKinnon has lost his fight against extradition in a case which has become a cause célèbre for civil liberty campaigners.
Both Mr Cameron and Nick Clegg, the Deputy Prime Minister, spoke out against extradition while in opposition but the failure to deport Mr McKinnon, who is accused of hacking into Pentagon networks, has recently threatened to sour Britain's relationship with the US.
The Government is thought to have sought a compromise agreement in which Mr McKinnon, 46, would be tried in America with any sentence served in this country.
Yesterday, Mrs May said she would like to see British criminals convicted abroad transferred to prisons in this country "as quickly as possible".
Mr Burrowes increased the pressure on the Prime Minister last night by telling friends he will resign as a member of the Government if Mr McKinnon, who has Asperger's syndrome, is deported.
A friend of the Conservative MP said: "He can't have blood on his hands. The mood music is not good."
His intervention came amid warnings that Mr Cameron and Mr Clegg will be accused of betrayal if they deport the computer hacker. Mr Cameron had said he was "deeply saddened" about the case.
Karen Todner, Mr McKinnon's lawyer, said last night that she hoped the Prime Minister and his deputy would honour the promises they made.
She added that psychiatrists had agreed that Mr McKinnon would be at high risk of attempting suicide if he were transferred to the US for trial.