According to Dom Consentino, the New York Jets are expected to look into Matt Schaub of the Houston Texans as part of their plan to shore up the quarterback position. Schaub has been a longtime starter for the Texans but is expected to be cut, mostly because he's been terrible lately.
As Consentino notes, Schaub has said he doesn't want to be a backup, so that may preclude him from further discussion. Even if he is not interested, the Jets should not be interested, in my opinion. Schaub is not very good, as he actually set the record for most consecutive games with a pick-six (four) this season. If Mark Sanchez is good at throwing pick-sixes, Schaub is the master.
I also expect Schaub to cost a pretty penny, just because he's better suited to step into a game situation if the need arises better than most other backups, simply due to his experience. As a result, I would not pursue Schaub if I were John Idzik. But alas, I am not. So we shall see.