The Paraguayan does not need surgery after suffering the injury in last Saturday's 0-0 draw with Swansea.
Ahead of this Sunday's trip to Sunderland, Toffees boss Roberto Martinez said Alcaraz is set to return to action next month.
He said: "We decided not to go through surgery.
"I think the specialist feels that is the best way to approach the injury, which is great news - it reduces the period that Antolin is going to be out for.
"We will find out over the next few days how he responds, but probably now we are talking three, four, five weeks maximum, which is fantastic news given what we feared earlier."
Martinez also said full-back Seamus Coleman was a doubt for Sunday's match after he suffered a cut to his Achilles tendon in the Swansea game that needed stitches.
Coleman was not part of the squad for Thursday's 3-0 win over Lille in the Europa League, during which the Toffees suffered another injury setback.
Midfielder Gareth Barry came off with an ankle problem and cut shin and he will also be assessed prior to the Sunderland game.
Martinez said: "This morning Gareth had a really long cut on his shin, but that is nothing to really worry about - the only worry is the ankle.
"He took a massive knock and we need to give it probably another 24 hours."
Meanwhile, winger Kevin Mirallas is expected to be back in action after his hamstring injury in "two or three weeks".