The 34-year-old Spain international defender had arthroscopic surgery on the cartilage in his right knee and the surgeons Ramon Cugat and Ricard Pruna refused to name a date for his possible comeback.
However, Puyol is almost certain to miss both legs of the Champions League quarter-final against Paris St Germain in early April.
"He will set his own recovery time. It could be a month-and-a-half, two or three," said Cugat.
"With a normal person you would say leave it a few months until it is better; with Puyol it could be four, six or eight weeks."
Pruna said: "He has been carrying this problem since October, and the game at Getafe. It hasn't gone away.
"He said he couldn't stand it anymore, and he wanted to use the break for internationals to try and be in top shape for the end of the season.
"If he didn't do it now there was a risk it could get worse.
"There is a chance he could be back in time for the Champions League final on May 25. We would all be pleased if that were the case because it would mean we had reached the final."