The former world number one, who had his hip resurfaced in January, is due to play doubles at Queen’s Club next week.
Murray will partner Feliciano Lopez at the Fever-Tree Championships in London.
But the 32-year-old says he still has “quite a lot of work to do” before he can be competitive in singles again.
“It’s baby steps just now,” Murray said.
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“I’m feeling good, pretty much pain free and enjoying just training, practising, improving all the time just now.
“I don’t think when Wimbledon finishes that I will just step on to the singles court the following week and everything’s good.
“At this moment I’ve been progressing all of the time. At some stage it’s probably going to plateau for a while before I’m able to kick on.
“I hope at some stage this year I would be able to get back to playing singles again. I’m not really interested in putting a time limit on it because I’m quite happy just now.
“So I don’t need to play singles after Wimbledon or at the US Open to – you know, enjoy doing what I’m doing just now.
“If I can, that would be brilliant, but I don’t think that’s going to be the case. I think it’s going to take a bit longer.”
No player has ever resumed a singles career after the operation Murray had, although American doubles player Bob Bryan returned five months after the surgery in 2018.
Murray has had a few singles practice sessions since being given the all clear to increase his workload three weeks ago. One of those practices was with this year’s Australian Open quarter-finalist Frances Tiafoe at The All England Club.
Murray is avoiding extended rallies for the moment, but hopes to be able to focus more on singles after Wimbledon, where he is expecting to compete in the doubles.
“I have done some singles training drills with my coach,” Murray said.
“Earlier on I was hitting with singles players but I was more stationary. I was moving them rather than them moving me about.
“So I have not played proper singles. I am hoping that will come more after Wimbledon.”
The WTA has confirmed that Amazon has won the exclusive rights to broadcast the women’s tour for the next four years.
Amazon’s online channel Amazon Prime, which already has rights to 39 ATP men’s events, will broadcast a minimum of 49 tournaments in 2020.