Rodgers had a major influence on Brown during his time at Parkhead and is said to have reached out to his former captain following his Pittodrie exit to discuss a possible coaching role within the Foxes' youth set-up.
The Daily Record claim an invite has been made for Brown to travel to the King Power Stadium over the coming weeks to spend some time with Rodgers and the Leicester coaching staff.
Brown is said to have had numerous offers since departing the Dons with Celtic also believed to be open to a return for one of their most decorated players of all time in an academy-level coaching capacity.
Brown brought the curtain down on his Celtic career last summer when he joined Stephen Glass at Aberdeen on a two-year player-coach deal, but his coaching responsibilites were removed when Glass was sacked last month and replaced by Jim Goodwin.
That prompted the former Scotland international to quit the club in order to pursue coaching opportunities elsewhere with a view to becoming a first-team manager in his own right.
Brown is now weighing up his future with his family before deciding his next move.