The report claims both Chelsea and City remain hopeful of clinching a deal for the Uruguayan, but are preparing contingency plans should he become prohibitively expensive.
And Germany international Gomez is the back-up target. City are thought to be leading the chase for Cavani, and the report claims the loser in the battle to sign the Napoli star will look to Gomez instead.
Bayern are believed to be willing to listen to offers for the 27-year-old striker, who's fallen behind Mario Mandzukic in the pecking order at the Allianz Arena.
Gomez's agent Uli Ferber said: "Mario is not happy with his situation."
The Germany star has made just seven starts in the league for the Bundesliga champions this season, although he's scored eight goals. Mandzukic leads the club's scoring charts with 15, while Gomez has been forced to compete with Claudio Pizarro for the back-up role.
It's a far cry from last season when Gomez was the club's top scorer with 26 league goals.
Both Chelsea and City are thought to be in the market for a new striker this summer. Chelsea signed Demba Ba from Newcastle in January, but neither the Senegalese nor Fernando Torres have managed to hit top form at Stamford Bridge yet.
City sold Mario Balotelli earlier this season, and with speculation mounting over the future of Bosnian forward Edin Dzeko, Roberto Mancini could be tempted back into the transfer market at the end of the season.