According to a report in the Daily Mail, the Belgian club has tabled an offer worth £13million to sign the Kenyan captain from North London.
The British newspaper further reported a top club from Serie A has also expressed interest in Wanyama as Brugge are yet to agree the fee or personal terms for the 28-year-old.
Brugge have offered £9m plus £4m in add ons and Tottenham anticipate another similar offer from Italy.
Wanyama is eligible for his British passport this summer but is eager to leave Tottenham to kick start his career after falling down the pecking order under Mauricio Pochettino following a few niggling knee injuries.
Wanyama, who is yet to feature for Tottenham this season, started his professional career with Belgian side Beerschot but is interested in the challenge of playing in Serie A where his brother McDonald Mariga represented Parma and Inter Milan.
With the addition of Tanguy Ndombele and existing midfield resources of Moussa Sissoko, Harry Winks and Eric Dier, Mauricio Pochettino has plenty of options in the middle of the pitch.
Wanyama still has two years remaining on his current Tottenham contract.