The German champions have reportedly been granted permission to speak to Sane, who is yet to sign a new contract with the Premier League champions.
Should the Bavarians place all of their interest in Sane, it would pave the way for the Red Devils to make their move.
Manchester City would prefer to keep Sane but will not allow his contract to run down and face him leaving for free so would rather sell sooner if he does not agree to a new deal.
In addition to City’s failure to present a financial offer that matches Sane’s judgment of his own worth, the young German is unhappy with his status in Pep Guardiola’s squad after being left out of the starting line-up on several occasions.
Bayern themselves are in need of a major overhaul after falling behind in the Champions League, and according to the Daily Record, Sane has been highlighted.
A timid loss to Liverpool in the last-16 did not go down well with club bosses and with Franck Ribery and Arjen Robben set to leave this summer, new faces are needed.
Bayern’s hierarchy has already signalled its intent to spend heavily on a summer rebuild by agreeing a club-record transfer fee of £68m for Atletico Madrid defender Lucas Hernandez, while taking fellow France World Cup winner Benjamin Pavard from VfB Stuttgart.
Bayern have already had a £52m bid for Pepe rejected by Lille, who have said that they will sell the star but want around £70m for him.
Pepe, 23, was one of several players former manager Jose Mourinho proposed to Manchester United’s board ahead of last summer’s transfer window.
Current boss Ole Gunnar Solskjaer has a huge rebuilding job on his hands this summer after another season of disappointment for the club.
United were unable to secure Champions League football for next season and may struggle to attract Europe’s best.
But, according to the Daily Record, a senior United executive contacted Lille last week to discuss the cost of acquiring Pepe and the likelihood of securing a deal.
Though a formal offer has yet to be made, Lille are aware that the English club has prioritised the signing of a right winger.
It remains to be seen whether or not Sane decides to put pen to paper on a new deal at Manchester City now that the Premier League season is over.