Recent Rangers signing Barton, 33, was quoted as saying Brown was "nowhere near his level".
"I've not been in his league, he's in my league," joked Scotland skipper Brown, 30.
"I think we'll get on really well, him being a Celtic fan, so I think we'll get on great."
Meanwhile, Brown is relishing playing under new manager Brendan Rodgers.
"He's been really positive with what he's done on the pitch so far," said Brown.
"It's exciting everybody, the way we're going to play and we're just looking forward to every single moment of it.
"We're going to be a lot closer on and off the field so it's going to be good times for us ahead. It's just a positive vibe throughout."
Of Celtic's league opener against Hearts on 6 August, the captain said: "We've got a new manager, we've got a fresh start. We'll show what we can do."
Brown was missing injured for a period midway through last season and then sat out the end of the campaign after Celtic had secured a fifth straight title.
"That's me back to normality," he added. "I've had a good four or five weeks off for the first time in 10 years so it's been lovely.
"I feel fine. I feel just as fit as I was last season when I was playing injury-free."
Celtic captain Scott Brown expects to get on well with Rangers midfielder Joey Barton, saying the Englishman is a fan of the Premiership champions.