I'd say it with absolute confidence right now, unless something very, very odd happens at one of the current top four clubs then there is no way that Liverpool are making the Champions League next season if Brendan Rodgers is the manager. They're more likely to slip behind Spurs and Southampton.
Real Madrid is the big one for Klopp. There aren't tons of high profile candidates that are potential managers for them - given that Simeone won't cross the city, not directly anyway - and it would go against Perez's history to retain Ancelotti after the season has fizzled out. Of course he might have been told that he's not in the running there and given that it seems that Pellegrini is getting another year at City (not that City's current squad is in any way compatible with Klopp's football) then there aren't any other really big superclub jobs going to come up this summer.
If he isn't getting the Real gig then he might be tempted by Liverpool, I don't like them as a club but what with their history they are still a big pull, also the ego boost you'd get as a manager if you were the one to finally win them the league again. And you're probably not expected to go and win the title straight away, you'd have time, you just need to show consistent progress. He wants to manage in England now his English has improved a lot over the last couple of years. If it were in any way possible then FSG would be mad not to go for it, Rodgers -> Klopp is an upgrade on a par with Saunders -> Jackett, they're not even in the same stratosphere.
I can't believe they're sitting there as owners of a club for whom they have aspirations of rejoining the elite, then they're looking at Rodgers continually dicking around with formations, three at the back, no strikers in the team despite him signing four of them, wide forwards at wingback, his transfer record as a whole, listening to the utter shit he comes out with and thinking he's the man for the job.