Brown returned to the international fold for the 3-0 defeat to England at Wembley less than three months after quitting.
However, he is swithering about seeing out the World Cup qualifying campaign with the Scots due to face Slovenia on March 26 – with a warm-up friendly fixture against Canada now confirmed for four days earlier at Easter Road.
England will arrive at Hampden on June 10 – which is likely to be just over a week before Celtic would be due back for pre-season training if – as looks certain – they are involved in the Champions League qualifiers.
Rodgers, who could have as many as six players involved in the Auld Enemy clash, admitted the situation is far from ideal.
“He’s at that stage now where he has to look after himself. I’m sure he’ll make the decision in time.
“It’s a difficult one. An international manager thinks about the now, but you’ve got to think about the future.
“We all understood the reasons why Scott came back to play in the England game.
“Going forward, he has some big decisions to make.
“He’s clear in his own mind how he sees it evolving, but, like I say, my worry is for Celtic and I want to make sure he’s the best he can possibly be for us.
“We’ve talked. He’s pretty clear on where I see him at and I’ve had experiences of these types of situations.
“I remember at Liverpool with Steven Gerrard and he was at this stage with England, as well.
“It has to be right for Scott. So, he knows where he needs to be at, in order to play the game at the level he wants to be at.
“He’s in a different stage of his life and career now. He knows that.
“Of course, I can give him my aspect on it and my feelings on it. He will speak to Gordon, I’m sure.”