The broadcaster, citing unnamed sources, said the last-place Twins -- who are in a rebuilding mode -- floated Mauer's name on the revocable waivers list Monday but didn't get any takers.
Mauer, 29, has a long-term contract on which he is still owed $142.5 million and also has a no-trade clause, meaning he would have to agree to any potential trade.
Fox reported it was notable Mauer wasn't claimed by the Boston Red Sox, who last week shed more than $270 million in salary commitments when they traded stars Adrian Gonzalez, Josh Beckett and Carl Crawford to the Los Angeles Dodgers.
The broadcaster said it indicated Red Sox GM Ben Cherington is apparently committed to a newly declared "discipline" in acquiring players with hefty price tags.
Mauer, the 2009 American League MVP, is a five-time all-star and three-time AL batting champion. He was slowed by injuries last year but has rebounded to hit .309 with 66 RBI and nine homers this year while playing catcher, first base and designated hitter. He is a .322 lifetime hitter.