A Mets official disputed to the New York Post a similar report by Fox's Ken Rosenthal.
Balfour chose the Rays because they were closer to his Clearwater, Fla., home, a source told ESPNNewYork.com.
Balfour went 1-3 with a 2.59 ERA and 38 saves in 65 relief appearances for the Oakland Athletics last season.
The Mets' offer suggests they do have payroll flexibility under the right circumstances. The payroll currently stands at about $87 million.
The offer also underscores how, despite the optimism, it is not fully known how Bobby Parnell will recover from herniated disk surgery.