The Brazilian international has been pushed down the pecking order at the Allianz Arena following the arrival of Javi Martinez last summer.
And the recent purchase of Barcelona starlet Thiago Alcantara could see Gustavo deemed surplus to requirements.
Mikel has been linked with a move to Turkey ever since Jose Mourinho re-joined the club as manager last month.
Galatasaray were thought to be suitors - with Didier Drogba reportedly persuading his old teammate to join him at the club.
And the report claims Mourinho has already picked out his replacement for the Nigerian international in the form of Gustavo.
But the Brazil international is believed to be keen to stay with the European champions. He recently played at the Confederations Cup with Brazil and his hoping to feature in Luiz Felipe's Scolari squad for the World Cup next summer.
As a result, Mourinho hopes to appeal to Gustavo's desire for regular first-team football. The competition at Stamford Bridge will be less fierce than at Bayern - especially if Mikel moves on.
Marco Van Ginkel has been brought in already, but the 20-year-old doesn't fulfil the same midfield role as Gustavo.
The report claims Gustavo would cost at least £16m considering he still has two years left on his current contract.
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