Connolly would at best be a short-term solution to a long-term problem. Connolly has started at guard and center for the Patriots since the 2010 season, but at 33 he's not exactly a spring chicken. He's also struggled to stay healthy for full season, having only one season with 16 starts under his belt. And to make matters worse, Pro Football Focus graded him as one of the very worst guards in the entire NFL last year.
If he signs with the Bucs, they may be able to count on him for a year or two, or he may see father time catch up with him -- in which case the Bucs would have to turn to disasters-waiting-to-happen like Oniel Cousins, or completely unproven and inexperienced players like second-year fifth-round pick Kadeem Edwards.
Connolly was with the Patriots when Bucs general manager Jason Licht and director of player personnel Jon Robinson were there, so that would explain some of the interested. Plus, he's one of the very few starting-capable offensive linemen still available. Or at least, theoretically starting-capable.
The Bucs were also reportedly interested in Stefen Wisniewski, who hasn't signed with a team yet. Tampa Bay managed to pass up the opportunity to sign any free agent, and then released Anthony Collins, leaving them with two big holes on the offensive line at tackle and guard. Signing Connolly would give them a little extra competition at one of those positions, but I'm not sure I'd rely on him to start.