Spain forward Villa fractured the tibia in his left leg in Thursday's Club World Cup semi-final in Japan and will be sidelined for up to five months. Netherlands midfielder Afellay had knee surgery in October and is targeting a return around the beginning of April.
"We have discussed it with [sporting director Andoni] Zubizarreta and the coaching staff and the squad is closed," Barca vice-president Josep Bartomeu told Spanish radio.
"Although Villa and Afellay are injured we have a sporting philosophy that tells us we have to fall back on the youth team," he added.
"The players from the academy, when they have played, have given really great results."
Villa arrived back in Barcelona on Friday and after a series of tests it was decided to perform an operation on the fractured leg on Monday, Barca said on their website.
Surgeons were planning to brief the media after the procedure, the club added.
The 30-year-old Villa has said he intends to return in time to play for Barca in May's Champions League final, should the holders qualify, and for Spain in the European Championship starting in June.
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