The 33-year-old midfielder, who has been only offered a one-year deal by the promoted Clarets, is in talks with the Scottish powerhouse over a possible three-year deal.
Miller revealed that he ran into Barton at a golf day and the duo discussed the potential move in detail.
"It was a great day for it and we had a good chat on the way around.
"It's no secret that the club are trying to persuade him to come to the club and hopefully we've done enough.
"The manager has never talked down the size of the job here and the size of the club so it would be a big thing if we could bring a player of that quality to this football club.
"He's at the stage of his career where he has the potential to play Premier League football and if we can get that signing over the line, it would be a massive coup for the club.
"He seemed genuinely interested and keen to know what the club was about. He knows the size of Rangers but he seemed interested in the subtle things about the club too."