The midfielder will remain a Reds player until the end of July before his six-year Anfield career comes to a close.
The deal gives the 32-year-old the chance to be involved in Liverpool’s long-anticipated Premier League title success and say farewell to his team-mates and other colleagues.
“I am really pleased to have the chance to finish this season – it means a lot to me and my family,” he said. ”Of course, one of the big personal positives about committing to this short-term extension is that I will now get to say goodbye to the people who mean so much to me in the right way.
“The gaffer has been different class and we’ve had an honest conversation about what my role will be in the remaining weeks. I want to contribute in the most positive way possible that helps him and the team, not just for the remaining matches of this season – but also what they’ll need to consider for next.
“I’m totally at peace with what that looks and feels like, and I know the role I’ll be playing for this team will help in what’s going to be an extraordinary period – the likes of which none of us have ever experienced.
“I feel in my time at Liverpool I’ve always looked to do the right thing for the team, the manager and the club in every decision I’ve made. I think this one fits into that category as well, so I’m buzzing to be seeing out this campaign and hopefully get to celebrate something memorable with an amazing group of people.”
Liverpool are six points away from clinching their first title in 30 years ahead of their return to action following the coronavirus suspension against Everton on June 21.
Manager Klopp said: “I know how highly sought after Adam is in this moment by other clubs who want him for next season, so for him this is a commitment to us and we are very appreciative to him for it.
“This is absolutely how it should be, too, because I am so pleased he will be with us to finish the job this season and have the farewell he deserves so much.”
The German added: “Adam always puts the club first – always. In the conversations we have had, he has been the first to express that my priority as a manager needs to be the players who are at this club for longer than just the rest of this season.
“He wants to be here to finish the job and do it properly, but it says everything about him that his biggest concern was blocking the progress and opportunity of a player who is under contract for next season also.
“But his role is too important to lose him before this season is done. So, it’s fantastic we can delay our messages of gratitude and good luck until the end of July now.”