The artwork shows three 1950s-style agents, wearing brown trench coats and trilby hats, using devices to tap into conversations at a telephone box.
It appeared overnight on a street in Cheltenham, Gloucestershire, just a few miles from GCHQ, where the UK's surveillance network is based.
Residents say they saw a group of men packing away a large white tarpaulin at around 7.30 on Sunday morning, before driving off in a maintenance van.
The celebrated Bristol street artist has yet to officially confirm the piece on his website.
Karen Smith, 48, said the figures appeared on the side of the house where she lives on Sunday morning.
"It's pretty good. It livens the street up a bit," said the mother-of-five. "There have been a lot of people about today looking at it.
"My daughter, Sophie thinks it's Banksy, but I've been speaking to different people outside and some agree, some don't."
Graffiti website streetartnews said Banksy had showed up with a van and covered the phonebox with tarpaulin before creating the piece "at the break of dawn".
"Quite a strong statement against the recent privacy issues we experienced this past year with the NSA and such," the website said.
"Banksy didn't pick Cheltenham randomly - the city is home of the GCHQ.
"This piece has all the Banksy characteristics in narrative, style and execution."
A spokesman for GCHQ said: "Although we are not qualified critics, we are as intrigued as the rest of the residents of Cheltenham about the artwork.
"For those who are interested, our website (www.gchq.gov.uk) gives a glimpse of what modern day intelligence operatives are really like, although some may be disappointed by the lack of trench coats and dark glasses."