A Jennings fight would help Parker boost his profile in the US before a potential unification bout against WBC champion Deontay Wilder next year. And Jennings, an experienced and credible boxer, is promoted by Top Rank's Bob Arum, who has a business relationship with Parker's promoters Duco Events.
"We have some very big options on the table for next year," Barry said. "Now, more than ever, we can't take our foot off the gas in terms of training. Joe is a world champion - we have to keep him in shape. We have to make sure all the niggles are taken care of. We've had some very awkward opponents. The last fight [Hughie Fury] was the most awkward guy in the heavyweight division."
Japan's Kyotaro Fujimoto, Australian Lucas Browne and Russian Alexander Povetkin were all potential opponents for Parker following his majority points win over mandatory challenger Fury in Manchester last month, but none of the deals stacked up for Parker and his team. The decision means Parker has fought only twice this year - by far his least active as a professional.
Barry said he envisaged Parker fighting at least two voluntary title defences next year. While all three recognised world heavyweight champions - Parker, Anthony Joshua and Wilder - said they were eager to unify the division, money remains a major sticking point.
Joshua's promoter Eddie Hearn has said Parker is in line to fight Joshua next year should the Briton get past Carlos Takam today, but that's news to Parker's promoter David Higgins.
"I haven't talked to him [Hearn]," Higgins said. "He sends me random emails occasionally but seems to do most of his announcements through the media."
Meanwhile, Takam, who has taken his fight against Joshua in Cardiff on 12 days' notice, has spoken of his poor preparation before his points defeat to Parker in Auckland last year.
"When I lose a fight, I accept that I lost it," Takam said in Cardiff. "Against Parker, I respect the decision of the judges. I try not to look at it as a defeat. It's just a way to learn more. I always ask myself what I could have done better in the fight. Also, I should have arrived earlier in New Zealand to acclimatise. I got tired in that fight."
Takam arrived in Auckland from Paris 11 days before the fight.