Mourinho, who looks set to take the reins at Chelsea following the announcement that he will be leaving Real Madrid, is keen to add another striker to the squad.
Moves for Atletico Madrid star Radamel Falcao and Napoli's Edinson Cavani have been mentioned, but both forwards are in-demand.
Chelsea face competition from Manchester City and Monaco for Falcao, while City are thought to have opened talks with Napoli over Cavani.
Gomez, who's found his playing time limited at the Allianz Arena this season, could be available at the end of the season.
The Germany international may be surplus to requirements once former-Barcelona boss Pep Guardiola takes over at Bayern, having already seen Mario Mandzukic usurp his place in the starting XI this term.
The Bavarian giants have also been linked with Borussia Dortmund star Robert Lewandowski - a move which would almost certainly end Gomez's hopes of nailing down a first-team place.
Gomez's agent, Uli Ferber, confirmed his player has not been happy with the situation at the club, and the report claims he'd welcome the opportunity to move on.
A fee in the region of £25m would reportedly be necessary to bring Bayern to the negotiating table. But Mourinho would be backed in the transfer market by Roman Abramovich should he return for a second stint at Stamofrd Bridge.
His three-year reign at the Bernabeu will come to a close next month, after Real Madrid president Florentino Perez announced the two will part ways on Monday evening.
Madrid will also waive Mourinho's £12m compensation clause, allowing him to re-join the Blues for free.
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