"I can't think of anyone who deserves one more," said Mr Cameron, who watched the match from the Royal Box.
But Murray said: "It's a nice thing to have or be offered but I don't know if it merits that."
David Cameron: "Honours are decided independently but frankly I can't think of anyone who deserves one more"
Mr Cameron said Murray's win had "lifted the spirits of the whole country".
"It was a fantastic day for Andy Murray, for British tennis and for Britain," he added.
"We were wondering on Sunday morning, 'do we dare to dream that this is possible?' and he proved absolutely that it was."
The prime minister cannot personally award an honour to the 26-year-old, stressing that they are decided independently.
People are nominated for their achievements by members of the public and whether they are given an honour - and the type of honour - is decided by a committee.
The committee's decision goes to the prime minister and then to the Queen.
was awarded an OBE
in the 2013 New Year's Honours list after winning Olympic gold and the US Open last year.
On the possibility of receiving a knighthood, the Scot added: "I think just because everyone's waited for such a long time for this [winning at Wimbledon], that's probably why it will be suggested."
In the first of several interviews on Monday, Murray told BBC Breakfast that he had only slept for "about one hour" after his victory.
Recent sporting knights and dames
Mary Peters, Kelly Holmes, Tanni-Grey Thompson, Sebastian Coe
Chris Hoy, Sarah Storey, Bradley Wiggins
Trevor Brooking, Bobby Robson
Stirling Moss, Jackie Stewart
Steve Redgrave, Matthew Pinsent
Clive Woodward, Ian McGeechan
Ellen MacArthur, Ben Ainslie
* Awarded since 2000
"You don't want to go to sleep in case you wake up and it hasn't actually happened," he said.
Murray later took part in a question and answer session on social networking site Twitter, ahead of attending promotional events for his sponsors. Murray then met Mr Cameron at a special reception at Downing Street shortly after 16:00 BST.
He was also greeted by a group of politicians including Deputy Prime Minister Nick Clegg, Labour leader Ed Miliband and Angus Robertson from the Scottish National Party.
After meeting the Wimbledon champion, Mr Clegg tweeted: "Pleasure to meet @andy_murray today. He's a true British champion who made tennis history yesterday."
And Mr Miliband told his followers: "Fantastic to meet @Andy_Murray and see the famous Wimbledon trophy up close after yesterday's amazing win."
Mr Robertson also took to Twitter to say: "Great to have the chance to congratulate @andy_murray personally on his @wimbledon victory."
"He's come into his own and there's a lot to look forward to," said the American.
Murray's win against Djokovic was watched by a peak audience of more than 17m viewers on BBC One.
The final also dominated social networking sites, with Twitter saying eight of the UK's top 10 trending topics as Murray hit the winning shot were Wimbledon-related.
At around the same time - 17:25 BST - the number of mentions of the final peaked at 120,000 tweets per minute.
Murray also added more than 230,000 new fans to his Facebook page, up to a total of nearly 1.3 million likes, during the second week of the tournament.
The Dunblane-raised player dedicated the triumph, which earned him prize money of £1.6m, to his coach Ivan Lendl. The American-Czech, 53, won eight Grand Slam titles as a player in the 1980s and 1990s, but failed to win the Wimbledon men's singles.
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