The Germany international had previously been heavily linked with Italian side Juventus, with some reports claiming that personal terms had been agreed over a summer move.
Juve's general director this week admitted that the club are doing "everything possible" to sign the player, but it appears that they may have to fight off rival interest in order to do so.
Can is out of contract at the end of the season and as such can discuss a pre-contract with overseas sides.
Italian publication 'Tuttosport' now claims that Juventus face late competition for the German, with Bayern Munich having entered the chase.
Should Bayern steal Can's signature it would be a case of rubbing salt into the wound for Liverpool, who also missed out on Schalke midfielder Leon Goretzka to the Bavarian giants.
Premier League leaders Manchester City are also said to be monitoring the situation.