German news magazine Der Spiegel claims the Prime Minister issued the warning to German Chancellor Angela Merkel at the EU leaders’ summit in Brussels on Tuesday.
Quoting sources close to those at the summit, the magazine said Mr Cameron fears the appointment of continuity candidate Mr Juncker would destabilise the UK Government to such an extent that it would bring forward an in-out EU referendum.
Mr Cameron reportedly added: “A figure from the Eighties cannot resolve the problems of the next five years.”
A Downing Street spokesman said: “We are not commenting on this. It was a private meeting.”
Mr Cameron has previously made it clear he views the former Luxembourg leader as a symbol of Europe’s past and has argued that a reformer should take charge of the EU executive.
However, on Friday Ms Merkel gave her backing to Mr Juncker for the key post, putting her and the Prime Minister on a collision course. Both Mrs Merkel and Mr Juncker’s parties are members of the European People’s party (EPP) bloc, which still dominates the parliament and has selected him as its preferred candidate.