The Cube, which must find ??185,000 before 31 December when its leasehold expires, intends to auction the print.
The work comes from the "Di-faced tenner" series - a set of spoof ??10 notes replacing the Queen's head with an image of the late Princess of Wales.
The signed and authenticated framed print was one of 50 produced.
It will be valued later in the week, a Cube spokesman said. One was sold at auction for ??24,000 in 2007.
"We have one of only 50 signed prints in existence that feature multiple 'uncut' notes with both sides visible," he added.
Once the freehold is bought, the Cube added, they would continue improvements to the building.
"This means we'll be continuing our fundraising activities. All proceeds from the 'Di-faced tenners' auction will go towards this ongoing goal, which is appropriate, as it's rumoured that Banksy frequented the Cube in his pre-fame days," he said.
The Cube is run by about 150 voluntary staff who have been renting the building since 1998.
The venue, which stages left-field film showings and performance art, is funded with income from the bar and the door.
Its 15-year lease expires at the end of the year and volunteers said they "jumped at the chance" when the landlord offered to sell them the building - if they could raise the funds in time.
Their supporters include Arts Council England who have pledged ??90,000, and other established venues in the city have also helped.
A recent benefit gig at the Bristol Old Vic raised ??5,000 and the Cube's neighbour ArtSpace LifeSpace "chipped in ??1,000".
The rest of the money raised has come from the "generosity of the public".
Cube volunteer David Hopkinson said: "It's very exciting to have the piece donated.
"Obviously the nature of anonymity means we don't know who it's from.
"But it's fitting as both Banksy and the Cube emerged on the Bristol independent art scene at the same time and artwork to do with money often sells particularly well in auction."
A rare Banksy artwork has been donated anonymously to an independently-run arts venue in Bristol which is campaigning to buy its building.