Murray defeated Serbia’s Novak Djokovic 6-4 7-5 6-4 on Sunday to delight a country that has waited so long for a successor to Fred Perry.
The world number two, who also holds the US Open and Olympic titles, has quickly been installed as the 1/20 favourite to win the Sports Personality of the Year award but there is also speculation he could earn recognition in the New Year honours list.
While Cameron has no influence over what is an independent process, the Conservative Prime Minster believes Murray is deserving of being knighted, saying on Monday: "Frankly, I can't think of anyone who deserves one more.”
Cameron, who watched Murray’s victory from the royal box at Wimbledon, added that the moment of sporting glory "lifted the spirits of the whole country".
He added: "It was a fantastic day for Andy Murray, for British tennis and for Britain … We were wondering yesterday morning, 'Do we dare to dream that this is possible?' and he proved absolutely that it was."