BR wants to manage another 'big six' team and he knows that won't be Liverpool. Any opportunity he gets to go to one of the others, he will be on the motorway in a heartbeat.
His career plan (which he has discussed in interviews) is to prove himself at the top level in the PL and then manage overseas (Spain) and then go into international management. He takes his career extremely seriously as we saw when he ditched Celtic so easily. He sees us as a stepping stone to get a bigger job and progress that career plan. He will trust his ability to make any of the big sides successful irrespective of what state they are in currently.
If Chelsea want BR, he will go there. You don't go from Leicester to Real Madrid/Barcelona, you only go to that sort of job from one of the big PL sides.
BR is currently 4th favourite to be the next Chelsea and Man Utd manager, 2nd favourite to be the next Arsenal manager, and favourite to be the next Man City manager.
He's going to one of these clubs at some point. It's just a case of when.