Murray made his singles comeback on Monday night at the Cincinnati Masters against Richard Gasquet.
The two-time Wimbledon champion gave a good account of himself and showed some promising signs that he is heading in the right direction, but ultimately the Frenchman had too much for Murray.
The 32-year-old admitted after the match that he felt good about his progression after undergoing hip surgery back in January.
Federer was in the Australian Open dressing room after Murray broke down in pain against Roberto Bautista Agut.
Murray’s injury issues had clearly taken a toll on him physically and mentally.
Federer knows himself better than most what it’s like coming back from injury problems in the past having won the 2017 Australian Open following a spell away from the sport.
And the 38-year-old, who is also in action this week in Cincinnati, was thrilled to see Murray continue his road to recovery.
“I saw him in the locker room after his match with Bautista Agut in Australia and he didn’t look good,” Federer told Eurosport.
“He was confused as to where his career was going to go, I remember that.
“People weren’t sure if he was going to retire or not, and he’s had major surgery now, and he’s back playing the doubles and now the singles for the first time, so I’m very happy for Andy.
“I hope he’s going to return to his old strength.
“I know it’s a long long way to go, and surgery wasn’t all that long ago, so it’s super encouraging and positive that he’s willing to take it to the next level now.”
Federer gets his Cincinnati Masters tournament underway tonight when he faces Argentine Juan Ignacio Londero.
The pair have never met previously but Londero has enjoyed a breakout year on the tour.
The 25-year-old reached the fourth round of the French Open where he lost to eventual champion Rafael Nadal.
Londero began this week with an impressive straight sets win over Matteo Berrettini but his opponent tonight is aiming to win a record 8th title in Cincinnati.
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