0551 GMT June 26, 2022
The 35-year-old Briton, who won the grasscourt Grand Slam in 2013 and 2016, sustained the injury during his Stuttgart Open final loss to Matteo Berrettini on June 12, Reuters reported.
The former world number one withdrew from last week's Queen's Club Championships and is racing to be fit for Wimbledon, which begins on June 27.
"The plan is still to play. The injury is improving. I've had it rescanned and it's progressing in the right way," Murray told British media on Monday.
"But you can probably work out from the nature of the injury which shots it is that I've been struggling with and haven't been able to practise.
"In the next few days I'll start to increase that and test it out. Hopefully it's recovered sufficiently."
Murray, who is now ranked 51st in the world, looked in good form during the grasscourt swing, reaching the Surbiton Challenger semi-finals before the final at Stuttgart.
"It's frustrating but I played nine matches in two weeks, which I haven't done since 2016," added Murray, who has endured lengthy injury lay-offs in recent years due to hip surgeries.
"My game was obviously in a good place physically.
"It's frustrating to be in this position in the build-up to Wimbledon, but the positive for me is I've dealt with certainly worse issues in the last few years and handled them OK."