Parker, the current WBO champion, is being pursued by Bellew following the Liverpool fighter’s win over David Haye, although Barry says a unification with WBC ruler Deontay Wilder is also a possibility.
“Here’s a guy that pulled off an amazing upset in Tony Bellew calling us out,” Barry exclusively told World Boxing News.
“The injury to David Haye obviously changed the fight in his favor but we were excited anyway because Tony was calling Joe out after winning.
“We have a lot of options though and we spent a lot of time at the last Deontay Wilder fight. We were down there with his team, both before and after the fight.
“They are very keen to get something on with us and we seem to have a relationship with them.
“So we know there are some opportunities in the future. It’s very exciting times in the heavyweight division and we are excited to be part of it.
“Joe loves his role as heavyweight champion and has loads of respect for those who were champions before him. He’s proud to wear the belt,“ he added.
Hughie Fury has solid claims to being Parker’s next opponent as the mandatory challenger, although a ruling is yet to be made on whether that bout has to happen in the immediate future.
With Bellew hoping to gazump Fury and Wilder having a purse bid with old-foe Bermane Stiverne to deal with, Parker’s options could be slimmer by next month ahead of a possible return to action over the summer.
Phil Jay is Editor of World Boxing News. Follow on Twitter @PhilDJay