Vertonghen was substituted 15 minutes from time at Selhurst Park, having fallen awkwardly following a clash with Palace striker Connor Wickham.
On Monday, the Football Association charged Wickham with violent conduct in relation to the incident, where he appeared to swing an elbow at Vertonghen.
The Belgium international is now set for a spell on the sidelines, though the club haven't specified how long he will be out for.
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"Our medical staff will continue to monitor his progress during his rehabilitation to determine when he will be ready to return to training.
"Get well soon, Jan!"
Vertonghen also posted an upbeat message on Twitter as he prepares to begin his recovery.
Happy to be here with the team! Thx for all your messages & support! Feeling positive about my recovery— Jan Vertonghen (@JanVertonghen) January 26, 2016