Goal understands the Croatia international has turned down interest from Arsenal and Tottenham to complete the move to Merseyside.
The centre-back was outspoken earlier in July after revealing Southampton had turned down a similar bid without informing him but has now been confirmed as Liverpool's fifth summer signing.
"It's a dream come true. I'm happy and glad to be here. I'm proud and can't wait to start. I know we have millions and millions of supporters - Liverpool is more than a club," he told the club's official website.
"After I played at Anfield last season, I said to myself: 'One day I hope I'll be able to play here for Liverpool'.
"The fans will know from the first minute that I'm a player that will give 100 per cent on the pitch."
Lovren joins former Southampton team-mates Rickie Lambert and Adam Lallana at Anfield, and new manager Brendan Rodgers has confirmed the defender will link up with the squad on their US tour.
"This is a very important signing for us and I'm delighted it's completed so Dejan can join up with us on our US tour," Rodgers added.
"He is a commanding and powerful presence and clearly has leadership skills, which is important; he fits the profile of player we are looking for.
"He is still relatively young, so his peak years are ahead of him and I believe he will improve and progress even further with us."
Lovren is the latest name on an ever-expanding list of players leaving Southampton, following Maurico Pochettino - who left for Tottenham - out of the door/
Luke Shaw has joined Manchester United for £30m, while Morgan Schneiderlin is the subject of interest from Tottenham and Arsenal and Calum Chambers is expected to join the Gunners in a £16m deal.