"We're 100 per cent confident that ... Joshua will turn that fight down," Lonergan said. "Takam is very experienced and very tough.
"We think they will protect their position because they are making big money selling out 22,000 at the O2 Arena [in London] and I've heard they are getting 400,000 pay per view buys. There's no need for him to go for the world title just yet."
Lonergan expected to hear within two weeks whether the 24-year-old Parker was guaranteed a shot to become the mandatory contender on or before January 16 next year.
"The mandatory spot is everything. No [champion] is going to fight Joseph Parker voluntarily. One, because we come from a small market that doesn't bring much money to the table and, two, we're dangerous."
Parker's next fight will be in Christchurch in late April before another in New Zealand - venue yet to be announced - in July. Neither of Parker's opponents has been confirmed, but Lonergan said they would be a step up.
There was talk about the Christchurch fight being held in the afternoon to cater for the American TV market, but Lonergan said it was possible they might instead hold it at the usual time late at night to best promote it on British TV.
"In England, boxing is hot ... but we think Joseph Parker could reinvigorate the heavyweight division in America ... in a perfect world I'd love to see Joseph Parker become the house fighter at the MGM Grand and that's something we'd like to move towards in the near future."
Fury, Joshua, Dereck Chisora and the return of David Haye have boosted the sport in Britain, with Fury easily the most dangerous.
"Anthony Joshua is a myth and [promoter] Eddie Hearn is going to protect him for as long as he can," Lonergan said of the undefeated 26-year-old. "I see one day when we do fight Anthony Joshua. The only thing that will hold us back from a three-fight trilogy will be the fact that Joseph Parker knocks him out in the first two or three rounds and no one cares about the second fight."
Lonergan said Parker would fight five times in the next 12 months, a schedule he would keep to should he win a world title.
"When we win the heavyweight title - and I've spoken to [trainer] Kevin [Barry] about this - we will be fighting five times a year. Our goal will be to win the division, whether it's the WBO or IBF, but also to unify the title."
Australian Solomon Haumono recently called out Parker, saying he wasn't as good as he believed he was, with Lonergan saying: "Solomon is an interesting one, he's very durable. But Solomon's problem is he's 40 years old."