Minnesota will instead exercise a $2 million buyout, which makes the veteran right-hander a free agent.
Minnesota, though, can still retain Nathan by signing him to a new deal, and general manager Bill Smith said the Twins are interested in bringing Nathan back.
"I spoke with Joe and his agent this morning, and expressed our interest in re-signing [him]," Smith said. "We will remain in contact with them as we move forward into the free-agent process."
Nathan, who turns 37 next month, posted a 4.84 ERA with 13 saves in 48 appearances totaling 44 2/3 innings last season. He pitched much better after a stint on the disabled list that began in late May after suffering a setback related to the Tommy John surgery that kept him out of action in 2010.
After making his return on June 25, Nathan posted a 3.38 ERA, with 28 strikeouts and 11 walks in 31 appearances over 29 1/3 innings, while converting 11 of 12 save opportunities.
The 11-year veteran is the Twins' all-time leader with 260 saves in his seven seasons in Minnesota. He picked up save No. 255 on Aug. 10 to notch the club record, passing Rick Aguilera, who set the previous record with 254 saves from 1989-98.