At a press conference at the start of the Shanghai Masters, reported by the BBC, the Swiss champion offered his appraisal of the British player's chances to take the top spot:
"He [Murray] has results that back up his chances to become World No. 1, maybe even at the end of the year, maybe at the beginning of the next year. If not then, he's got a shot till next year's Wimbledon almost if he were to win there.
"His next nine months are going to be extremely interesting to follow.
"I think he's done so well, I said it right off the bat, his reaction was amazing right after not winning Wimbledon against me, then coming back and winning the Olympics, ... then bringing the victory home, his first Grand Slam, at the US Open. [I’m] very impressed, [it was] great to see. I hope for him he can achieve it eventually.”
Roger Federer regained the No. 1 position this summer when he won Wimbledon for the seventh time, defeating Murray in a four-set final. That victory also allowed the Swiss champion to tie and later surpass Pete Sampras's record of 286 weeks as world No. 1.
As of today, Federer has held the top ranking for a record 299 weeks.
If Andy Murray does take over the No. 1 spot, he will be the first British man to occupy it since the current ATP system began in 1973. Murray's previous high was the No. 2 ranking he achieved in 2009.
Murray fought hard at Wimbledon and gave an emotional speech after his loss, breaking down in front of the capacity crowd, many of whom were hoping for the first homegrown men's winner in 76 years.
Federer notched up his 17th grand slam victory and said of Murray, "I really do believe deep down that he will win grand slams and not just one.”
Murray may have lost the match, but he won the hearts of the British public with his all-out efforts. Previously some of the crowds had been lukewarm to the Scot’s achievements.
Giving a heartfelt message after that loss, he left many of the British fans in tears, and said he was getting closer. A month later on the same court he succeeded, beating Federer in straight sets to take Olympic gold, leaving the Swiss with silver.
Federer has the men's doubles gold medal he won partnering compatriot Stanislas Wawrinka at Beijing in 2008. Having won just about everything else a player could hope for in tennis, Olympic singles gold was one honor that eluded him.
Despite his graciousness in defeat, it must have been a bitter disappointment to come so close to a prize that only comes around every four years.
Gracious in victory and defeat is one of Federer's trademarks.
Rather than harbor resentment against the Scot, who denied him perhaps his best chance at that prize, he has given a boost to Murray's profile as well as his confidence with his latest comments.
Will Andy Murray become Men's Tennis No. 1?
Murray was praised by defending US Open champion Djokovic after Murray triumphed at the tournament this summer, where he battled both the weather and Djokovic. Nadal, absent from the tournament, sent him a congratulatory message. Murray has finally achieved what the tennis elite thought he was capable of achieving.
The strange thing is it has taken a long time for the fans to warm up to Andy Murray, but he earned the respect of other players some time ago.
Scottish nobility come out to see him and the British public is finally behind him.
Now Roger Federer says he can take the top spot.
The question is, will he?