The Sunday People claims that the British PM is taking a "massive interest" in the potential move south of the border for the Glasgow giants.
"Two major powerbrokers involved in the process confirmed that meetings have taken place recently with Cameron and other leading figures," the paper reports.
"And the Prime Minister’s team are planning to pull strings to make the move happen sooner rather than later."
A "prominent figure, who wishes to remain anonymous" claims that the two clubs have "never had a better chance" to earn a money-spinning move to England.
Cameron is apparently trying to persuade the FA and the Football League to allow Celtic and Rangers to jump straight into the Championship, thus avoiding the potential logistical nightmare of lower league sides seeing their often tiny grounds inundated with tens of thousands of Rangers and Celtic fans heading to away matches.
In addition, Cameron apparently wants to force a deal through before the Scottish independence vote in September 2014 - since he believes that it could be a key factor in stopping Scotland breaking away from the United Kingdom.
"Rangers fans are squarely behind the unionist cause and will always wave the flag. But if Celtic were to be playing in England it might prove to their fans that they should be British," the paper's source says.
"This may sound strange to people in England but there are hundreds of thousands of votes here – and football is at the middle of a lot of debates up here."