The Turkey international was a target for several Premier League clubs after struggling to make an impact at Real last season following his arrival from Borussia Dortmund.
Arsenal had shown an interest in the 23-year-old, but Sahin has tied up a switch to Liverpool after successfully coming through a medical.
He is looking forward to the challenge of making an impact at Anfield and is hoping he can cap his year on Merseyside by winning a trophy.
“I am happy to be here and I am looking forward to playing for Liverpool,” Sahin told the club’s official website.
“After the first conversation we had, I was left impressed by Brendan Rodgers. He told me about how he wants to play football and what the goals are he wants to reach with Liverpool.”
“It makes me proud to have joined such a big club like Liverpool Football Club.
“After the first conversation we had, I was left impressed by Brendan Rodgers. He told me about how he wants to play football and what the goals are he wants to reach with Liverpool.
“I am looking forward to meeting the fans. I hope it will be a nice year with trophies.”
Liverpool boss Brendan Rodgers is confident that Sahin, who was the Bundesliga’s Player of the Year in 2010/11, will fit into his style of play.
Rodgers said: “We are delighted to welcome Nuri to Liverpool and also to say thank you to Real Madrid for loaning us such an exciting talent.
“Nuri Sahin is a player with great technique and motivation and someone who will fit in smoothly to our playing model.
“We welcome Nuri into the Liverpool family and wish him good luck in his efforts to make us successful in our future.”