If all goes to plan three-time grand slam victor Murray hopes to be back in singles action later this year, although he has put no time frame on his return.
Britain's Andy Murray plays a backhand return during his men's singles match against Spain's Roberto Bautista Agut on day one of the Australian Open tennis tournament in Melbourne on January 14, 2019.
"It's baby steps just now", Murray told reporters of his return back to tennis.
He said he was "pretty much pain-free" now and enjoying training, including some singles practice sessions.
"I don't think when Wimbledon finishes that I will just step onto the singles court the following week and everything's good".
"At this moment I've been progressing all of the time".
"When that is, I'm not really interested in putting a time limit on because I'm quite happy just now so I don't need to play singles after Wimbledon or at the US Open".
"[Against Tiafoe], I did a lot of the usual drills I would do if I am going to play singles but I didn't play a set with him".
Murray underwent hip surgery in January, soon after a tearful news conference at the Australian Open when he said the tournament could be his last after nearly two years of injury problems.
However, Murray still expects to take his time before returning to singles action, which probably rules out competing at the US Open at the end of August.
Asked whether he had been playing singles in practice, Murray replied that he had not gone so far as to play through full rallies and sets with top-class opposition.
Murray added: "I haven't played singles practise sets". I have done some singles training drills with my coach.
"So not necessarily doubles specialists, but guys that if I was able to would be up for playing, but likewise if I'm not, they're fine as well".
As for the possibility of an on-court collision with Jamie at Queen's, Andy said: "I think mentally for me if I was playing singles it's a bit more hard. I did first four shots in the rally, a couple more, but no extended rallies".
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