Unless the Klopp news has been leaked to throw everybody off the scent then it seems as good as done. There's been no "x will take over 1st team responsibilities until we appoint a successor" line out of the club and they've said they expect to have a new manager in time for the Spurs game.
There's been reports tonight that Rodgers fate was decided as long ago as the Utd game. The last few weeks have been used to talk to possible replacements and now's the perfect time tie up the details and give the new manager time to settle in before his first game.