Sun on Sunday reported that Bayern officials are under the impression that they have a better chance of signing the England international than Manchester United, who had an approach rebuffed earlier this summer.
Dier can play as a defensive midfielder, a centre-back or a right-back and has four years left on his Spurs contract.
His sale could also provide Spurs with a much-needed windfall, with the building cost of their new stadium predicted to soar past £800m – double the initial estimate.
However, Spurs may be reluctant to sanction a deal for Dier after having sold Kyle Walker to Manchester City for £50m. The north London club are the only side in England's top four divisions yet to make a summer signing.
Spurs boss Mauricio Pochettino is reported to be interested in signing Everton midfielder Ross Barkley, who has one year left on his contract and has failed to sign a new deal.
The Toffees are holding out for a fee of around £50m for Barkley, although that figure is likely to drop towards the end of the transfer window.
"We made a really good offer to him to sign a new contract, he declined that contract and told me that he is looking for a new challenge," Everton boss Ronald Koeman was quoted as saying by the Guardian last month.
"He is not in Everton's future any more. It is his decision. I need to respect that and see what happens. What I heard from the board at the moment is that there is no offer on the table for Ross.
"It is really clear. I am looking to other players. They are my priority and not Ross, with all respect, because it is his decision. Everybody knows what the situation is."