But Bremner said he had no idea he had helped to thwart a rebellion 27 years ago.
He told the BBC: “You think you’re being satirical but the reality is even more farcical.
“Honestly, who’d have thought that without a spoof phone call, a group of Eurosceptic backbenchers might have forced the prime minister to resign and taken us out of the EU. What are the chances of that?”
One of the rebels, Sir Richard Body, believed he had been called by the then-prime minister, who had a slim majority of 18 and was constantly fending off criticism from his party’s Eurosceptic wing.
Sir Richard told the Cabinet Secretary Sir Robin Butler that “if it had been an actor he would have been at the very top of his profession”.
He was so convinced that he called fellow MP, John Carlisle, who had also received a call and they decided to call off their challenge.
Bremner made the calls as part of his new show but Sir Richard was insistent it had been the real prime minister and told the cabinet secretary the call had “saved [Sir John’s] bacon”.