Protective hoardings were removed and holes made at the four corners of the artwork "Spy Booth".
Businessman Hekmat Kaveh, who has agreed to pay an undisclosed amount to keep the artwork in place, said he had arranged for a restorer to look at the artwork, to see if it could be saved.
Two weeks ago, a Perspex cover was put up to protect the artwork, after it was daubed with silver writing.
The piece, depicting men "snooping" on a phone box, appeared in April, three miles from government listening post GCHQ.
Mr Kaveh said: "There are four very large holes on the four corners, which I've measured, and one is 10cm.
"It looks as if it was being prepared to be taken away, despite the fact I'm in the process of purchasing it to make sure it stays.
"There's been damage to the plaster and the painting.
"It's been reported to the police and the council. It's on a listed building, it's criminal damage."
In a statement, Gloucestershire Police said it was called at about 13:00 BST on Monday "to investigate a reported offence at the Banksy site in Fairview Road Cheltenham".
It continued: "Unknown offenders removed a piece of wood surrounding the Banksy artwork on the side of the building and a number of holes were drilled around the image.
"The artwork was not damaged and there is no sign of forced entry to the property. "
Campaigners have been trying to keep the artwork in situ on the corner of Fairview Road and Hewlett Road, after the owners of the house claimed it had been sold and workmen arrived to remove it.
Last month, Cheltenham Borough Council issued a temporary stop notice preventing further removal work from taking place on the Grade II* listed building.
Mike Redman, the council's director of environmental and regulatory services, said: ''As the property is listed, the removal of the wall, or a section of the wall, requires listed building consent (LBC)."
The Banksy mural on the wall of a house in Cheltenham has been attacked again.