Carlo Ancelotti is thought to believe that Dier could fancy a move to Germany, after a deal to take him to Manchester United fell through.
Dier was an influential part of Tottenham's season, with his 36 Premier League appearances helping them achieve a second place finish last campaign.
If Dier were to leave he would be another high profile player to depart Mauricio Pochettino's side this summer, after Kyle Walker left for Manchester City.
Spurs have already come under fire for not replacing Kyle Walker, such that it would be a huge decision by Daniel Levy to also let Dier go, if a replacement is not already lined up.
But with reports claiming that Dier may fancy Munich, after the United move collapsed, then Spurs could sanction the sale and look for a replacement.
The only rumoured replacement for Eric Dier mentioned so far has been Inter Milan's Geoffrey Kondogbia, however Pochettino could look to re-invest in other areas of the pitch.
Tottenham are said to have been tracking Everton midfielder, Ross Barkley, but up to now have been put off by his £50m price-tag. Dier's potential sale to Bayern would effectively mean Spurs are back in the running for the out-of-favour Everton man.
Tottenham have yet to make any signings this summer and a right back, to cover Kieran Trippier, is still a priority.
In order to fill that position Tottenham have been linked with both Benjamin Henrichs of Bayer Leverkusen, as well as Jeremy Toljan of Hoffenheim, both of whom have been valued at around £14m.