Brown is one of the best midfielders in the Scottish Premiership, but the former Newcastle United and Queens Park Rangers man told talkSPORT that the Scotland international is not in his league.
Barton also took a dig at new Celtic manager Brendan Rodgers, highlighting to The Mirror the fact that the former Liverpool and Swansea City boss has not won anything.
Former Celtic midfielder Grant believes that it was crazy of Barton to make such comments about Brown and Rodgers.
“To come out with stuff about Scott Brown and Brendan Rodgers before a ball has been kicked is crazy,” Grant told The Glasgow Evening Times.
“We all know that Joey likes to be forthright with his opinions but I don’t think he quite realises what he is setting himself up for. He would be far better to shut his mouth and do his talking on the pitch."
Barton’s comments are controversial, and they have spiced things up ahead of the 2016-17 Scottish Premiership campaign.
Rangers won the Scottish Championship title this past season and will feature in the Scottish top flight during the 2016-17 campaign.
Celtic will be favourites to win the league title next season, but Mark Warburton’s side will be expected by many to give the Bhoys a run for their money.