They likely do have a deadline, or, hopefully, back up plans in place that they've duscussed internally. I don't see a stated deadline working out in the Twins' favor. Especially since they already know that they'll have to get cteative to sign him and with other teams interested already (soon to be more once Judge signs). If they insist on a deadline, they'll have to put an offer out there that he can't refuse; which equates to overpaying.
I know the author is speculating, but I think that if the Twins feel like they're in that situation in which they need to set a deadline, they might as well pivot now and begin talks with Bogaerts.
What seems to be moving in their favor, and I can only speculate based on the tea leaves left by the various articles that I have read, is that teams once floated as highly intetested (i.e. the Cubs) seem to be shifting to other shortstops. (Again, if the reports are to be believed.) This could indicate that they know that the Twins are serious about resigning him and they'd rather move on than get in a bidding war. Or it could be that they don't want to wait and have pivotted already (meaning they're ahead of the Twins on that front). It could also mean that they want to get in front of the post-Judge market and sign players before teams miss out on Judge and look to Correa.
All I know is how I'll feel as a Twins fan if they're left out in the rain on adding a key piece -- hint: frustrated and disappointed. I'm sure management knows that this sentiment likely represents the majority if the fanbase and will act accordingly. You can't rebrand, open a bunch of cap space, not sign a star and expect fans to show up. J.A. Happ, Dylan Bundy, Matt Shoemacher, Chris Archer and Kurt Suzuki type aquisitons won't be enough. Not this time.