The Lost Bass Project is appealing for information about what it describes as “the most important bass in history”.
McCartney bought the instrument for £30 ($38) in Hamburg, Germany, in 1961, but it disappeared eight years later.
The hunt began after McCartney urged manufacturers Höfner to track down his beloved instrument.
The bass features in The Beatles’ music of those years, including the hits Love Me Do and She Loves You.
Nick Wass is heading Höfner’s search project and has joined forces with two journalists in trying to solve the “greatest mystery in the history of rock and roll”.
He has collaborated extensively with McCartney and written a book about the missing Höfner 500/1 Violin Bass.
Wass told the BBC that the famous Beatle asked him about the guitar during a recent conversation – and that is how the campaign to find it began.
It is not clear what happened to the instrument, which was put away presumably after the Beatles finished filming Get Back in 1969, he said.
“It’s not clear where it was stored, who might have been there.
“For most people, they will remember it… it’s the bass that made the Beatles,” Wass said.
Husband and wife team Scott and Naomi Jones, both of whom have worked for BBC News, are helping with the search.
Scott became curious about the guitar’s fate after watching McCartney’s 2022 headline set at Glastonbury and approached Höfner – only to discover they were already having conversations about tracking it down after being urged to by their famous client.
He told the BBC: “Paul said to Höfner ‘surely if anyone can find this guitar, it’s you guys’, and that’s how it all came about.
“Now we’re working together on this. Nick has more technical knowledge about this guitar than anyone on the planet, and me and my Naomi are bringing some investigative skills.”
It’s anyone’s guess how much the guitar would fetch at auction but there are some precedents.
John Lennon’s stolen guitar sold for $2.4m (£1.9m) when it resurfaced half a century later, and the acoustic Kurt Cobain played during the iconic MTV Unplugged set sold for $6m (£4.76m).
McCartney’s era-defining bass would likely surpass both – but the Lost Bass Project team are clear that there is no commercial motivation for their search.
Jones said: “Höfner’s hunch is that someone will come forward purely on good will, and whoever has it probably doesn’t even realise what it is they’ve got.
“It would be nice if it could go on public display one day – and if the only way someone is going to come forward is to make some money from it, then so be it, because at least it would be found.
“But ultimately we’re just doing this to get Paul his guitar back. We know via Nick and Höfner that it’s what he’s always wanted.”
The project’s public appeal has been live for less than 48 hours but the team has already “received exciting new leads” they are following up.
How to spot McCartney’s bass
There are a couple of tell-tale signs any amateur sleuth should be aware of if trying to identify the bass.
A dead giveaway is the Höfner logo, which is written vertically on the headstock of the original model but was horizontal on later versions played by McCartney.
The missing bass also looked very different last time it was seen to how it did in older pictures, because it had to be renovated after extensive touring.
McCartney’s missing bass was given a darker paint job, had the pearl pickguard removed and had the two pickups mounted in a single piece of black wood.