The 39-year-old left Swansea City to replace Kenny Dalglish at Anfield in May after Rodgers led the Swans to a respectable 11th place in their first season in the Premier League.
Cole scored four goals in 31 appearances for Lille during his season-long loan to the Ligue 1 side last season before returning to Merseyside earlier this summer.
The West Ham United youth graduate worked with Rodgers when the Northern Irishman was a coach at Chelsea under José Mourinho, and Cole is excited about the prospect of playing under him again.
“When I was hearing about Liverpool looking at certain managers, I was praying it was going to be Brendan — not just because he’s not an unknown for me and I know his philosophy on football, but I think for the club the owners have made the right choice,” Cole told the Sun.
“He has just impressed everyone so much since he’s been here. That style is what I was brought up with at West Ham and Chelsea.
“For me, that is football. I love the style of play. At Lille, we played like that as well — getting the ball down and moving.”
The 30-year-old midfielder, who has two years left on his current deal at Anfield, has scored three goals in 32 appearances for Liverpool since moving from Chelsea on a free transfer in July 2010.
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