After he indicated he would be open to the idea of staying at Anfield beyond 2024, the Reds are looking to step up talks with the manager.
The German has two years left on his current deal and had previously suggested he could step down at the end of his contract, but he is now open to extending his stay.
One of Mike Gordon's priorities was to discuss a contract extension with the 54-year-old.
Kosicke sat a few seats away from Gordon, who is leading the talks, after positive noises from the camp of the German coach.
The 2-0 win over Villarreal at home puts them in firm control of reaching a third European final in four years, and they are on course to win an historic quadruple this season.
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The FA Cup final is next month and his side are a point behind Manchester City with five matches remaining, having lifted the League Cup in February.
In his seven years at the club, the German has won five trophies, including the 2019/20 title.
After struggling last season due to the COVID-19 epidemic, Klopp is back to his best this season.
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The next two summer windows are likely to focus on refreshing the squad after key players have already been renewed.
The input from the two assistants, who have managerial ambitions, is critical to the future of the club. If neither of them want to stay on, that could have a big impact on the decision.
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The close relationship between the ownership and the manager has meant that they haven't wanted to force or box him into a corner.
FSG accelerated negotiations after the German indicated that he would be open to remaining at Anfield beyond his current deal. After enduring a draining last season as a professional and as a person, and being involved in the process of building the next great team through long-term recruitment planning, Klopp has been revitalised this season and has been involved in the process of building the next great team.
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It has become apparent to those at the club that he is leaning towards sticking around to manage the evolutions. Mike Gordon, the FSG president and man in charge of these discussions, has an incredibly strong relationship with Klopp. He often stays at the manager's home when he goes to Merseyside and has previously drafted fresh terms after a casual chat with the 54-year-old.
This is how the six-year contract was signed by Klopp. The manager said that he could imagine being at the club for the rest of his career. A new deal was drawn up in a matter of hours.
This was the case again in the year 2019, when it was shown that he had changed his mind on having a sabbatical and was tied down to 2024.
It is the will of the Reds boss that will determine whether he commits to the club or not, as well as if his trusted lieutenants, Peter Krawietz andPep Lijnders, still have the ability to take the club forward. The former may have managerial ambitions of his own.
Sky Sports will show two rearranged games in May, one of which will be Wolves vs Manchester City.
With five games of the season to go, City are top of the league by one point, followed by the Reds.
Two of their rearranged games will be shown live on Sky Sports.
The game between Man City and Wolves will be played on May 11 at 8.15pm. The rearranged game will take place on Tuesday May 17 at 7.45pm.
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On Monday Night Football, Jamie Carragher analysed the strengths and weaknesses of both Manchester City andLiverpool and looked at the areas which could cause them to drop points in the title race.
The top two teams in the league are separated by a single point after they both won at the weekend.
Both sides will slip up between now and the end of the season as they have five games each left to play.
Sky Sports pundit said on Monday Night Football that he could see both of the teams dropping points.
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