St. Louis coach Jeff Fisher said at the owners meetings in Orlando that the Rams are interested in bringing Sanchez into the fold.
"I'd say that there is interest. I can't say how much. But there certainly would be interest,'' Fisher told Jim Corbett of USA Today. "I don't have a backup with experience on the roster right now.''
Bradford underwent ACL surgery in October, and the Rams' current backup is second-year man Austin Davis.
Rams offensive coordinator Brian Schottenheimer worked in the same position with the Jets from 2009 to 2011, when Sanchez was the starter.
"Brian had a good relationship with Mark,'' Fisher said. "He's just learning. He's been to the AFC Championship game a couple of times, so he's been there.''
Sanchez, 27, was 33-29 in four seasons as the Jets' starter, with four road playoff wins. He has 68 career touchdown passes and 69 interceptions.
Bradford was off to a hot start last season with 14 touchdowns and four interceptions before injuring his left knee in a Oct. 20 loss to the Panthers. SI WIRE: Bucs wide receiver Mike Williams stabbed in leg, brother suspected