Nathaniel Clyne |
Sky sources understand the Liverpool have paid an up-front fee of £10milions, which will rise to £12,5milions based on incentives reached by the player.
Rodgers has already brought in Milner, Ings, Bogdan, Gomez and Roberto Firmino to Merseyside.
Clyne's arrival will see him reunited with three of his former Southampton team-mates who left the south coast for Liverpool last summer Rickie Lambert, Adam Lallana and Dejan Lovren.
Rodgers has pinpointed 5-cap England international Clyne as a replacement for Glen Johnson, who has left the club after his contract expired.
He had the first part of his medical on Monday and after that was concluded, he has now pledged his future to the team which finished sixth in the Premier League last season.
Clyne said: "I'm delighted and it's a dream move for me to come here and play for such a big club with all of its history. I can't wait to get started.
"As soon as Liverpool showed their interest in me, I was happy and just wanted the deal to go through.
"The gaffer has shown a lot of interest in me and this is a place where I can improve as a player, I think. It's definitely a good opportunity to come here and try to fulfil my potential.
"I've played at Anfield before and know what the atmosphere is all about. It's a great atmosphere and will make me rise to the challenge. I'm really looking forward to it and hopefully it can push me on to putting on good performances."
Clyne says he is aiming to win silverware in his first season with the Reds.
He said: "If we get a good pre-season out of the way and start the season brightly, it'll be looking good. You can see we've got the players to do it.
"I'm sure with all the competitions we're going into this year, we'll be going into them looking to try and win them.
"That's where I want to be - I want to be competing for all the titles and trophies that are available.
"I've been progressing as a player but I'm fully confident I am good enough to play for a team like Liverpool. What I need to do now is show that.
"Hopefully we can have a good season and come out of it with some silverware."
The defender also revealed that former Southampton team-mate Lallana played a key role in his arrival at Anfield.
He added: "He said it's a good club - a big club with a huge fan base. He said the team is good and a good bunch of lads. He told me I'd be made to feel welcome if I joined," he added.
"I suppose it'll help me settle down with the team. Knowing a few of the players at the club is good. I can join in with the lads and I reckon I'll settle down easily."