Rodgers worked as a youth and reserve team boss at Chelsea until Mourinho departed the club in 2007 and has worked his way up the managerial hierarchy since having had spells at Watford, Reading and Swansea City.
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He told BBC Radio 5 Live: "Brendan was a young guy when he started working at Chelsea. He had lots of energy, lots of enthusiasm, very good ideas.
"He was learning but also showing his ideas, sharing things.
"I enjoyed very much working with Brendan and we keep in touch. I am not saying every day or every week, but we keep in touch a lot because it's the kind of feeling we want to keep forever.
"I want him to be a success, to keep doing things well. I am so happy they [he and new West Brom boss Steve Clarke] are on top."