According to former GM and current ESPN analyst Jim Bowden (subscription required), the Cardinals were one of 4 teams (at least) to check in on Twins closer Glen Perkins, only to be told no. The other teams were the Reds, Dodgers and Braves.
In addition, those 4 teams are monitoring the health of White Sox reliever Jesse Crain and he could end up being a more viable trade candidate.
Obviously, the Cardinals are seeking pitching, either in the form of a starter (such as Matt Garza) or a reliever. Adding another arm in the pen would allow the Cardinals to return Carlos Martinez to AAA to work as a starter, while adding a starter would allow the Cards to move Joe Kelly back to the pen and shift Martinez to AAA as well.
In addition, Bowden put together a list of players from last weekend's futures game who could be traded, which included Kolten Wong. The thinking here, of course, is that, with Matt Carpenter locked in at second and David Freese seemingly locked in at third, Wong could be used as trade bait. Its unlikely that he would be used to acquire Crain or Perkins, if he were available, but he could be used for a bigger fish, such as Matt Garza.
Bowden also provided his list of expendable players, which includes Jon Jay, who he sees as expendable due to Oscar Taveras waiting in the wings.
I don't think the Cardinals are going to move Jay before the trade deadline, as Taveras just recently returned from his ankle injury and is not ready to take over center field just yet. If the Cards were to trade Jay, I think it would be an offseason move.
Speaking of Taveras, from the "tell me something I don't know" department, Jason Churchill of ESPN list Taveras as one of the "untouchable prospects". This pretty much goes without saying. I cannot think of any scenario in which the Cardinals would part with Taveras. Even the old Jurickson Profar for Taveras rumors were a bit farfetched.
In any case, with the All Star game out of the way, things will start heating up on the rumor mill.