Swindon midfielder Louis Thompson has agreed a move to Norwich, while Southampton's Jack Stephens has joined Town on loan.
The 27-year-old Dane has made 177 appearances since signing from Brondby in 2006 and spoke of his delight about signing the extension, the length of which was not revealed.
"It means everything to me. This is where I want to be and I'm happy the club want me. I am really happy right now. I've never wanted to go anywhere else," he told the club's official website.
"I feel I am a part of this club - and I want to be a part of this club in the future.
"I like the way it is going right now with the style we're playing, and hopefully the results will come soon as well.
"The support from the fans has been unbelievable. The game when they were singing all the songs about me, it was difficult to focus on the game! It was such a good experience for me and I'd never experienced anything like it before.
"That is why I am so happy here. That for me is Liverpool Football Club.
"I feel I belong here and as long as the manager and the owners want me, I'll be here."