Vicario has appeared in every league game for Spurs this season, as they sit one point above the bottom three in the standings.
"The minor procedure... has been timed to have as minimal an impact on our season as possible," said Spurs, who will go into a three-week international break after Sunday's league game against Nottingham Forest.
Vicario was not included in Italy's squad for Thursday's World Cup play-off against Northern Ireland. He last appeared for Italy in a 2-0 win over Moldova in November.
Gianluigi Donnarumma was the Italy keeper in a 4-1 loss to Norway three days later that forced the team into the play-offs while Norway qualified for the World Cup.
"It is hoped that he (Vicario) could return to action within the next month," Spurs added.