With Vertonghen and Alderweireld at the heart of their defence, Tottenham have conceded fewer goals than any other team in the top flight this campaign and Pochettino will have to turn to Austrian Kevin Wimmer or England's Eric Dier for cover.
Although Spurs did not provide a time scale for Vertonghen's return, such injuries usually take an average of four to six weeks to heal.
"Our medical staff will continue to monitor (Vertonghen's) progress during his rehabilitation to determine when he will be ready to return to training," Spurs said in a statement on Twitter.
Wickham has been charged with violent conduct and faces a minimum three-match ban if found guilty of elbowing Vertonghen.
The incident during 3-1 win at Selhurst Park was missed by referee Martin Atkinson but caught on camera.
Spurs are fourth in the table, five points behind leaders Leicester City.
(Reporting by Simon Jennings in Bengaluru, editing by Ed Osmond)
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