I wouldn't be the least bit surprised if Bortles lands in New England this offseason. There's going to be a limited number of teams looking for QBs and the Pats are going to be a desirable landing spot for backups with Brady's contract situation/career winding down. For all the grief that Bortles gets, he still checks some worthwhile boxes: he's played well against the Pats (which they seem to care about when it comes to free agents), he has all the physical skills you'd want in a QB, he takes care of the football. I remember reading (maybe from Lombardi?) that the Pats really liked him coming out of college.
There's some chicken and egg stuff when it comes to Bortles. Jax has a very conservative offense so Bortles rarely gets a chance to let it rip. Do they have a conservative offense because Bortles isn't good enough to thrive in a "modern" NFL offense or are they conservative because Coughlin, Hacket, Marone et al haven't caught up to the times. Both? (Probably some of both). Still, a $10m/year short-term contract for Bortles would make a lot of sense for both sides.