The experienced attacker is widely expected to leave his current club on a Bosman move and could be on his way to Italy or Spain.
Milan are said to be interested in signing an inexpensive back-up for centre forward Zlatan Ibrahimovic, and according to Tuttosport, the 32-year-old Klose would fit perfectly into their plans.
As Kicker reports, Valencia are also keen to strengthen their attack in the summer transfer window and they see the veteran, whose contract expires at the Allianz Arena in the summer, as the right addition to their front line.
Bayern have yet to officially confirm that Klose will leave, although negotiations between player and club have reached a standstill. The striker is demanding a two-year contract, while die Roten are only willing to offer him a one-year deal.
Klose started his career with Homburg and also wore the jerseys of Kaiserslautern and Werder Bremen before joining Bayern Munich in the summer of 2007.
But the German international has struggled to secure regular first-team action at FCB, and looks set to leave on a Bosman.
Klose has been linked with clubs such as Borussia Dortmund, Tottenham Hotspur and Lazio in the past few months, but Milan and Valencia now appear to be the front-runners to secure his services.