I could see a swap of 1sts with the Jets provided they traded Wilkerson to the Bears in the deal. The Jets are at 20 and they might want Paxton Lynch there.
I think the Bears would do that in a hearbeat. More likely though would be us giving up pick 11 and maybe getting back the Jets second or third round pick (essentially a bigger trade down). Kind of like what the Jets did to us when we traded Brandon Marshall.
You are right we would have to sweeten the deal but I cant see the Bears coughing up a 2nd or 3rd in addition to a 1st round swap, but I could see the Bears coughing up the Pats 4th or even their own 4th instead