Unfortunately I think he's going to be franchised. I think we are more likely to trade for Adam Thielen
I don't really see many ways the Bears can franchise tag him to be honest.
They essentially have no cap space right now. They could create some by cutting some players (Kyle Fuller, Akiem Hicks, Jimmy Graham), but I think they'd like to keep most of those guys. Contract restructures could be in the works, but they'd need to a lot of work.
The franchise tag for a WR should be in the range of about $16.5M, which goes straight to the cap. So realistically, they're going to need to find a way to clear at least $20M in salary cap space in the next week or so in order to be able to use the tag. AND... that just gets them basically under the cap.
Plus, the only QB on their roster right now is Nick Foles, who I'm certain they don't plan on starting. Given that they're way too late in the draft to be in the market for a rookie upgrade (nor do I expect anybody in their FO/coaching staff has the desire to reset there), they're likely in the veteran market. If they're in on even like an Alex Smith or a Ryan Fitzpatrick type, they're still going to cost like $8-10M a year or possibly more, and they'll certainly be signing short (potentially one year) contracts. So that's mostly straight to the salary cap.
So basically, in order to add a veteran QB not in the Watson/Wilson class (which I doubt would happen) and to franchise Robinson, it'll probably cost them at least $25M in cap space, which they currently don't have.
They could probably clear about $10M or so by restructuring Khalil Mack, but then you're pushing the can down the road on a guy who already has a $26.6M cap hit this year. He would be north of $30M in 2022, with pretty much no chance to cut... at 31 years old.
They could look to extend Kyle Fuller and push his $20M cap hit down some, but he's got at least $9M in dead money and he'll be 30 after this season.
Robert Quinn could get restructured and create potentially another $7-8M or so, but similar to Mack, you're staring at a guy in his 30s with a $20M cap hit in future years that you'd probably like to move on from at some point.
Akiem Hicks is in the last year of his deal and costs $12M against the cap with little dead money, but I think they like him, so he may just play it out.
I think you'll have a pretty good idea what the Bears direction is in the next week or so. If you see extensions/restructures starting to come through, it means they're either "making a splash" at QB, or they're freeing up space to keep Robinson. If not, I don't see many ways they can franchise him.