The seven-time Grand Slam champion picked up the injury in the first round of the Barcelona Open earlier this month before withdrawing from the tournament.
Scans showed the issue was more serious than initially thought and he then skipped the Madrid Open.
"The next test will be crucial," Alcaraz told Spanish television channel TVE.
"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," the 22-year-old added.
Alcaraz, who was crowned Sportsman of the Year at the Laureus Awards on Monday, surrendered the world number one ranking to Jannik Sinner after losing to the Italian in the Monte Carlo Masters final days before his Barcelona opener.
The French Open will start from May 24 in Paris.
