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Both camps have told the New Zealand Herald that they are close to signing a deal that will see the Aucklanders, both ranked inside the WBO top 10 rankings, fight before Christmas.
Matchroom's ebullient promoter Eddie Hearn overnight told IFLtv that Parker was due to fight in New Zealand, ramping up speculation that a Fa deal was over the line.
"It will be our first show in New Zealand and we're planning a great fight for him," Hearn said. "Looks like it'll be in October. It'll be a major coup to get this over the line."
Hearn was likely speaking before he became aware that Auckland had gone back into lockdown for three days ending tomorrow, although it is expected that will be extended.
Parker's manager David Higgins said the latest Covid-19 developments probably meant "October was a bit too soon", but that there was a determination to stage the fight before Christmas.
"Yes, negotiations had stalled but there have been vigorous talks behind the scenes and the fight is a lot closer to reality than it was a couple of months ago," Higgins said.
"We've been in negotiations now for six months and that's how long it took to put the Tua-Cameron fight together and Parker versus Joshua, so it's nothing new in terms of that.
"I have to say they're [the Fa camp] still being a touch greedy but if we can close the gap a bit we'll have something to announce very shortly."
Higgins wouldn't say how far apart the camp's remained but the Herald understands that Fa is guaranteed a $500,000 purse and potentially a cut of pay-per-view on top of that.
Keddell, Fa's manager, also refused to discuss the minutiae of the financial arrangements but acknowledged that "we are very close".
"Lou DiBella [Fa's promoter] has been in continual talks over the past three weeks and we're hoping to make this fight happen."
Heavyweight boxing is a fickle sport with first round knockouts as likely as a 12-round borefest with everything else in between, but a Parker-Fa clash has the potential to be an instant classic.
Parker is an Auckland-born Samoan and Fa an Auckland-born Tongan, which brings immediate frisson to the clash.
They have a split history in amateur fights and at the moment are ranked No 2 and No 6 respectively in the WBO heavyweight division as challengers to Anthony Joshua's belt.
The 28-year-old Parker has 27 wins and two losses in his pro career, which culminated in a WBO title when he beat Andy Ruiz Jr in 2016, while the 30-year-old Fa has won all his 19 fights.
Both are members of the same Mormon church.
With Auckland being the largest Polynesian city in the world it makes perfect sense to host it in the City of Sails, but coronavirus might yet make that impossible. Queenstown has emerged as a potential venue, too.
"We're not close on signing off on a venue," Higgins said, "but we'll host it where it makes the most economic sense."
Parker has not fought since beating Shawndell Winters by fifth-round technical knockout in February, while Fa has been inactive even longer. He last fought in Utah last November, beating Devin Vargas by unanimous decision.