Rex strikes me as the kind of guy who would want to prove his critics wrong. Prove he can make it in Chicago, prove he can be a quality QB.
He knows the Bears brass likes him. He has to feel that he can beat out Orton in an open competition.
I also don't think anyone is going to give him starter money on a contract longer than a year or 2. I know there are other teams that need QB help, but I can't think of any team that is going to "go after" him.
All that being said, I don't see why you wouldn't test the FA market. Although, Parker is probably getting a pretty good inclination as to who is interested and at what value at the combine this week. Angelo said he wants players that want to be Bears. Maybe he's trying to show that he is one of those players.
I know Grossman was killed by Chicago fans and media last season, but he was killed nationally, not just here.