Sources close to the actor tell TMZ ... Johnny's upcoming U.S. shows with his band, The Hollywood Vampires, are getting postponed on doctor's orders ... as a result of a nagging lower-body injury.

We're told JD recently tweaked his ankle during his Cannes press run, and he stressed it further while trying to power through some recent music gigs ... when he was doing back-to-back Jeff Beck tribute shows. You'll recall ... he was front and center for those in the U.K.

Now, our sources tell us Depp's ankle is fractured and his medical team has instructed him not to fly for the foreseeable future -- which means the first 3 HV gigs are off the table.

Those initial shows were on the East Coast -- Manchester, NH, Boston and Bethel, NY. We're told Johnny and co. are trying to find a way to reschedule them toward the end of their tour -- which will begin to wrap up around late July ... when they'll be in Europe.

Our source tells us Johnny's disappointed over this, explaining ... "Johnny went from two full days of press appearances in Cannes, straight to London for rehearsals and then played two tribute shows in honor of his dear friend, the late great Jeff Beck."

They add, "His ankle was already bothering him at Cannes, and it got progressively worse, but he wanted to honor his commitments for 'Jeanne du Barry' and for Jeff. He’s devastated to postpone the shows, but his doctor believes flying could make the situation much worse. He plans to rest up so the European dates go off without a hitch." Those will start in June.

As we reported ... Johnny's been taking his health much more seriously amid his comeback -- with a lot of rest and clean eating -- and this move is just further evidence of that.

The last time he was in the U.S. was way back in November -- when he did a concert with Jeff Beck here in L.A. Since then, the guy's mostly been across the pond ... this after his W in his defamation case against Amber Heard.