Levein suggested Brown had got himself deliberately booked against Kilmarnock on Saturday to ensure he served a suspension before Celtic face Rangers on March 11.
Levein’s latest comments came after Brown had claimed that his yellow card was down to his former Scotland manager’s jibe that “everybody should get a little bit more protection from Scott Brown”.
Rodgers said: “He is a wee bit obsessed with Browny, isn’t he?
“I think it is a huge compliment for Browny that Craig keeps talking about him, if he does, especially as we don’t play them for a little while.
“Browny is a very competitive player. There is nothing really to answer on that.”
Asked how comfortable he was with other managers speaking about his players in public, he said: “It is not something that any manager likes so much. It is not something I do unless I am praising a player for his qualities. It is each to their own.”
The row started after Celtic’s recent midweek victory over Hearts when Levein hit out over Brown’s challenge on Harry Cochrane, which left the 16-year-old with a collarbone injury, and accused him of seeking out the midfielder in response to Celtic’s 4-0 defeat at Tynecastle in December.
After the Celtic midfielder was booked against Kilmarnock last week - which rules him out of this month’s visit of St Johnstone to Celtic Park - Brown retorted: “He has done his job, hasn’t he? I was booked with my first foul and I think that is exactly what Craig was looking to do.”
Brown will be free to play in Saturday’s William Hill Scottish Cup tie against Partick Thistle and Rodgers is confident his side will bounce back from the 1-0 Ladbrokes Premiership loss to Kilmarnock, which was only his second defeat by Scottish opposition since he took over at Celtic in 2016.
He said: “I was disappointed after the game, I thought we could have done more despite the pitch and everything else.
“But I have to defend the players as well, it is an horrendous surface to play on.
“I have said (that) when we have won there. It is one of those things that can happen.”