It makes very little sense for Rondo to do this.
Rondo's value can't be terribly high right now. If he signed an extension at this point, it's likely he'd be signing at below market value.
Also, there's the matter of Rondo's age and where the Celtics are at right now. Barring a Garnett + Allen style flurry of moves, the Celtics are at least a couple years away from having the pieces to be truly competitive. Rondo will likely be in his 30s before the Celtics are ready to make a run in the playoffs.
I think for Rondo it makes sense to wait things out. He can come back (hopefully) fully healthy next season. He can try and have the best (statistical) season of his career. He'll have leverage to pressure the Celtics to either make moves to put some talented, experienced pieces around him, or else trade him to a team with whom he is interested in signing long term.
There are only two reasons that come to mind for Rondo to sign an extension this summer, and neither seem characteristic of him:
One - he's worried that he might get injured again or that his value will only diminish moving forward.
Two - he really wants to stay in Boston regardless of the situation and he wants to give the team a home-town discount.
For the Celtics, it would make a lot of sense to trade Rondo this summer if he won't agree to a contract extension. I don't think it would be a good idea to head into next season with the cloud of Rondo's impending free agency hanging over the team for months on end.
I also think a factor for the Celtics has to be the cost of keeping Rondo, even if he is willing to re-sign. Rondo is probably worth between 12 and 15 million to the right team. I don't think he's worth that to the Celtics, who don't have any elite scorers on the roster, and probably won't get one in the draft in June.
It's also possible that Rondo will want to get the kind of money that Chris Paul, Deron Williams, and Russell Westbrook are getting, whereas in my opinion fair value for him is closer to what Stephen Curry and Ty Lawson are making (I agree with Dos that Curry is WAY underpaid). The team should under no circumstances give Rondo a deal that will give him over $15 million per year.