The world number two recently pulled out of the Barcelona Open as a result of the issue and explained that the next scan on the injury will be "crucial" in determining whether he participates at Roland Garros - where he has won the last two titles - when the clay-court Grand Slam gets underway next month.
Alcaraz told Spanish TV channel TVE: "The next test will be crucial.
"We've been trying to do everything we can do to make sure that this test goes well. I'm trying to be very patient. But we are good, we are just waiting a little bit.
"We have a few tests in the next few days and then we will see how the injury is, and what the next steps will be."
Alcaraz explained that he won't rush his recovery just to play at the French Open as he is more concerned about his long-term tennis future.
The seven-time major champion said: "I have a very long career ahead of me, with many years still to come. Forcing things at this Roland Garros could really harm me for future tournaments.
"We’ll see how it goes. As I said, I’d rather come back a little later but in good shape than come back too soon, rushing things and not at my best."
Former Spanish tennis star Feliciano Lopez recently expressed concern that the injury could rule Alcaraz out for an extended period.
He said: "I’ve had that injury myself. From what I’ve heard, it’s a very common injury in the world of tennis.
"I think his wrist tendon is a bit inflamed, I imagine, and I hope it’s not ruptured.
"Now it’s time to recover because it’s not a muscle strain from overuse, since he arrived tired from Monte Carlo and something like this can happen to you. There are many small tendons in the wrist.
"I was out for about two months. I don’t know what his injury is like, the extent of it. I was out for two months, but of course, when I got it, I could barely hold the racket until a few days after the match ended."