The 36-year-old terminated his Aberdeen contract less than one year into a two-year deal earlier this week as he looks to make progress with the transition from his playing to coaching career.
Brown was a key member of Stephen Glass’ coaching staff but those opportunities and responsibilities would not be the same under new boss Jim Goodwin.
The former Celtic captain is now weighing up his next move with Rodgers, who managed the player for two-and-a-half seasons hoping to get him down to Premier League side Leicester.
“We’re going to have a chat about organising for him to come down because he gave me so much in my time there and I want to try to be reciprocal to him,” the Northern Irishman told the Daily Record.
"He’s someone that my door will always be open to.”
Rodgers believes Brown has what it takes to be successful as a coach, whether it be at Celtic or elsewhere but reckons he should concentrate on gaining the experience and skills required to be a success.
“I’m pretty sure if Brownie really wants to do that then I think he will do it," he told the Record. “But it’s about building up the skill set to best enable him to become the manager of Celtic and deal with the pressures and the expectations he’ll then encounter.
“I have absolutely no doubt he can then go on to do really well as a manager or a coach. It’s just about waiting for what he sees as the right opportunity.