The jetpack was designed by Glenn Martin before the liquidation of the Martin Jetpack Aircraft Company earlier this year.
It was listed for sale on Trade Me by Skylark Asset Realisation with a $1 reserve. The current bid is $46,000.
The sale ends at 7.40pm on Tuesday.
According to the description, the jetpack crashed in a lake in late 2016 so cannot be flown.
Canterbury Museum director Anthony Wright told the Herald Martin developed his first jetpacks in his Christchurch garage.
"So the Martin jetpack story is a fantastic tale of Canterbury innovation in the mold of Richard Pearse's aircraft, Bill Hamilton's jetboats and John Britten's motorbikes.
"We want to acquire a jetpack to make sure we're able to tell that story in the future," he said.
"It might even inspire future inventors."
There appears to be a lot of interest in the jetpack, with a total of 112 bids made at 2pm on Monday.
Canterbury Museum was leading the way but has since been overtaken by one bidder.
Bidding on objects at auction is part of the job for the museum's curators, but it is always a nerve-wracking process, Wright said.
"Especially for high-profile objects like this one. We really hope we're able to bring this jetpack home for everyone to enjoy."
"As an early investor and former director and chief executive of Martin Aircraft Ltd, I'm keen to keep a jetpack in Canterbury," Lauder said on the page.
The jetpack, model P12 ZK-JMK, was powered by a custom-made 200hp V4 two-stroke petrol engine based on the Honda CR500.
According to the listing, it has light signs of wear and usage but is a "well-above-average example".
It is one of seven prototype Martin Jetpacks that were being sold after the liquidation of the company.
