In addition to all the big/not so big club stuff, I suspect there is a purely football reason Can doesn't feel fully settled and comfortable here. Imagine you're the manager and you need to assign positions for two midfielders: one is a strong tackler, but average passer, the other one is the opposite. You would stick the defensively stronger one at CDM and play the more skillful one further upfield, right? Most managers would do that as well, but Klopp generally does the opposite. That's kind of the essence of "gegenpressing" – the players who are better at winning the ball operate further upfield, so that they can use their skills in pressing and tackling to force turnovers of possession closer to the opposition goal ("pressing is the best playmaker"), while the better passers play deeper in order to deliver the ball into those areas near opposition goal as quickly and reliably as possible and with less pressure from the opposition.
This creates an unsolvable problem for Can. As a player he is generally better and happier playing at #6 (CDM) – but his incurable tendency to dwell on the ball instead of passing it off quickly and efficiently is the opposite of what Klopp requires from his deep-lying playmaker (I'm sure the coaching staff did their best to change that trait early on, but soon realized that certain things are not coachable). So after Lucas has lost the remainder of his pace and his starting place in midfield, Klopp didn't turn to Can, but instead moved Henderson – a weaker defender than Can, but a quicker passer – to CDM and in most games when we use the 4-3-3 system Can has been asked to play further up the field at #8. Can has admitted in the past that he doesn't enjoy playing there as much because this position requires more frequent sprints and stops, which puts more stress on his knees and generally wears him out – he is simply not physically built to be that kind of a midfielder, he is too powerful and not mobile enough to play this way. So we have an intrinsic conflict between the what's best for the player and for the manager; the player shows more of his best qualities at #6, but the manager prefers a different type of player in that position.
This is unfortunate, because in every other way Can has been a perfect Klopp player – alongside Firmino he was the first to buy into his style and coaching methods, dramatically turning his form around compared to the miserable last few months under Rodgers. Can scored the first goal of Klopp's reign and if I'm not mistaken his 100th as well. He was the one who instigated the comeback against Dortmund at home in Klopp's most memorable LFC game so far, scored twice against Hoffenheim to ensure our participation in the CL group stage and scored the winner with an overhead kick, no less, last season in a crucial game in the run-in to the top 4 – all of that while doing the team's best defensive work in midfield. Can has earned the manager's trust to start in most of the decisive games – but I can understand why he would think he will only realize the full extent of his potential in a team that gives him the #6 position and accepts his style rather than tries to change it.
You can understand Klopp as well. He has not been overly stubborn or tactically inflexible – when necessary he adjusted his style and system to find a way to make things work with Can at CDM or occasionally in a 2-man midfield. He clearly appreciates what Can brings to the club, but he should not have to change his whole philosophy to accommodate one player. I think the potential signing of Jorginho is designed to cut the Gordian knot of compromises and uncertainty brought about by Hendo's chronic condition and ensure that the kind of quick-passing style Klopp enjoyed with Dortmund, particularly when Sahin was playing, will be here for good. And Can will either have to adjust to this reality or find a place for himself elsewhere.