Glasgow Rangers boss Mark Warburton has brought the midfielder to Ibrox, and he has taken little time to create some controversy.
The 33-year-old has taken little time to endear himself to Gers supporters by previously stating that the Hoops skipper was not in the same league as the one-time England international, who has recently moved to Ibrox.
But Hay has warned that Barton's lack of knowledge of what playing for one of the Old Firm is all about could ultimately mean that his comments come back to haunt him.
"The one thing I would say is that there is a perception that you have been there and seen it all when you get to these twilight years of your career. It is not going to take too long before Barton realises that Glasgow is different to anywhere else, and I don't that say that lightly," he told the Evening Times.
"He comes from Liverpool which, like Glasgow, is a city who loves its football but unlike Liverpool, Glasgow is divided not on football lines but on something that runs deeper than that.
"So I wouldn't be too surprised if there is someone at Rangers who is quick to have a quiet word in Joey's ear. It is all very well having an opinion and being forthright with it but in my day we had a saying that it was one thing to say it and another to actually do it.
"Think you're better than Scott Brown? Show us, Joey."
With Scott Brown being a key part of the Bhoys squad that has now won five Scottish Premiership titles in succession, as well as several other honours, the former Queens Park Rangers man will have to make a considerable impact in the coming years if he hopes to prove his point.
Whether Barton truly believes the gulf in quality between the pair is as considerable as he suggests, the comments may have been made to motivate himself, as well as the rest of Mark Warburton's team to prove upon their top flight return that they deserve to share the stage with their bitter rivals.