The Germany international, whose current contract with the German giants is due to expire, is also a target for Serie A side Lazio.
Barclays Premier League duo Everton and Norwich have also been linked with the 32-year-old's services, but the Turkish club are hoping to steal a march on its rivals.
Klose has been offered a one-year extension by Bayern, but is understood to be keen on holding out for a two-year deal.
Trabzonspor chairman Sadri Sener believes the signing of Klose would be a real statement of intent as they prepare for Champions League action.
He told Turkish newspaper Superlignews: "We are making progress in the negotiations we have with (Miroslav) Klose.
"We are doing everything we can to see him join us. A striker of his calibre would be a great boost for us in our upcoming Champions League campaign."