Well, he didn't help himself with the early transfers, Downing in particular has been the huge flop which the owners will be asking an explanation for. Was the player really in terrible form all season? Did Kenny's system not bring out the best in him? Why was this?
Additionally in the FA cup final against Chelsea, why wasn't Carroll played from the start? He was influential in the next game in the PL and took Terry apart... Things like that do allow FSG to question him pretty closely. They seem to be using a more corporate model of business management, which is "This isn't working, tell me why not?", which can only be a good thing for the club in the long term.
The question has to be who replaces him if he does go (prob sports director, though may take a break), AVB wouldn't be a good guy to transition a "poor" Liverpool considering his tenure at Chelsea. I'd like to have Rafa back, as he may have played boring ****, but he made the team tough to beat.