According to the Daily Mail, the remarks, recorded at a party in Central London last week, was made by former Army colonel Patrick Mercer, who was also a former homeland security spokesman for the Tories.
In the recording, he said the Prime Minister would be ousted by next year.
Asked if backbenchers would launch a leadership coup, Mercer replied: “He'll go in the spring. He'll resign in the spring.”
The row comes four years after the Newark MP was forced to resign as the Tories' homeland security spokesman when he made comments on race in the Army which Cameron condemned as 'unacceptable'.
The outspoken politician, who went on to be a security adviser to Labour Prime Minister Gordon Brown, was among backbenchers who rebelled last month over a referendum on European Union membership.