Spurs had reached an agreement with the Serie A side which consisted of £25.7m plus Djed Spence on loan.
But German champions Bayern have made an improved offer of £26.7m as they look to steal him away from Ange Postecoglou.
According to Sky Sports, it is now up to Dragusin to decide which club he wants to join.
And transfer expert Fabrizio Romano understands Tottenham remain confident of signing the defender having “improved their own proposal overnight”.
Spurs are hoping to fly Dragusin to London to complete the transfer, with critical hours ahead for all parties.
Postecoglou is in desperate need of a centre-back to bolster his depleted squad.
Cristian Romero and Ben Davies are both sidelined with injuries, while Micky van de Ven is only just returning from a lengthy spell out.
And Eric Dier is set to join Bayern Munich after a decade in North London.
Dragusin made his professional debut for Juventus in 2020 but struggled to establish himself as a regular first team player.
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He was subsequently loaned out three times to Sampdoria, Salernitana and Genoa.
The defender completed a permanent move to Genoa last January and has since made 62 appearances.
Dragusin would be Tottenham’s second signing of the January window after they brought Timo Werner in on loan from RB Leipzig.