Free agents have to want to come to Minnesota. The Twins have not had much success luring top FA starters.
If the Twins actually make the highest bid, and it's turned down, they won't want to directly say so, but they have their favorite reporters to whom they can make off-the-record hints to that effect and get the message into the papers and other outlets.
They don't dare play that kind of PR game falsely, lest the player in question quietly say, "um, no," when asked to confirm.
And so, in turn, when we don't see these scoops published by the usual outlets, it's a pretty good indication that that didn't happen. Players aren't turning down our good money. They're turning down our inferior money.