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The two camps have been in talks about a possible mid-year fight in Auckland, which would have rivalled the David Tua v Shane Cameron "Fight of the Century" in 2009 for public interest.
Parker, a Kiwi with Samoan ancestry, is a former WBO world champion and Fa, a Kiwi with Tongan heritage, is unbeaten as a professional. They fought each other four times as amateurs for a 2-2 record.
Higgins told Sky Sports in the UK: "Fa is advised, or promoted by Lou DiBella, and my understanding is that... the money they want is ridiculous.
"We're talking 10 or 20 times his biggest purse. Parker would have been willing to fight for less than Fa [requested], and Parker is a former world champion. That's called pricing yourself out.
"Fa was looking for far too much for where they're at. I guess it's a mark of respect for Joseph Parker. They are worried about what would happen, frankly.
"Their camp had made noises about fighting Joseph Parker, but they should back up their words. We wouldn't see them getting nothing either. Fa would get his biggest pay-day by several multiples by fighting Parker, but their demands were stratospheric, given the current circumstances.
"I think let's just give it time and meanwhile we'll be assessing other options."
A fight between the pair would have filled Auckland's Spark Arena, for example, but given the effects of the coronavirus, which has shut down sport around the world, the bout would have been in doubt for the original slated date of mid-year.
However, New Zealand's response to the pandemic has been successful so far and it's quite possible professional sport will start here far quicker than anywhere else.
Precautionary tests of Parker and Fa, plus those on the undercard and the accompanying officials and entourages, could clear the way for an event, but packing thousands of spectators under a roof may be more problematic.