Monk has played for Swansea in all four divisions and was club captain when Rodgers was manager of the club between 2010 and 2012.
The Liverpool boss said: "I speak a lot with the senior players whatever team I'm at and obviously at the time I was there he was a magnificent captain for me.
"In a lot of our conversations, which were very open, I said to him that I thought he should start thinking about his coaching, two to three years before he finishes playing.
"He was always a bright lad, always very honest and loyal as a captain, but he was one I could see having a future in coaching and management if he wanted to go down that route.
"Of course it's probably come unexpectedly, especially being put into the Premier League, but he's looked very calm and seems to be enjoying it.
"The players have responded to him. He'll have gained respect because of his status as a player and he'll be looking to make that similar impact over time as a coach and manager.
"He's a great learner and always has been, and I think over the course of the coming games and months he'll get better."
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