This will actually be interesting...the bears have the cap space to make it happen, but Briggs doesn't have a lot of leverage here unless there's an option coming up in his contract that will force the team to negotiate.
The new CBA makes it much more costly for a player to hold out, IIRC, which means that if Briggs wants a fairly small raise (i.e. only a couple million), then he can blow through that entire raise by holding out next season. The player's leverage has been reduced here.
That said...the Bears will take care of him if he keeps this behind the scenes and deals with things appropriately.