The discovery was made on a  Ryanair flight which arrived late on Saturday night, forcing the closure of the airport for six hours.
As a result flights 12 flights were diverted and others were delayed.
Ryanair said the "written hoax bomb threat" was found during a routine inspection of the plane.
An airline spokesman apologised for any inconvenience adding: "Police carried out a security search and cleared the aircraft to return to service at 08.55 BST [on Sunday].
Avon and Somerset Police said an investigation was under way and said everyone aboard the Ryanair flight would be questioned.
Passenger Zara Walker was returning from Portugal on an easyJet flight just behind the Alicante plane. She said it eventually landed in Cardiff.
She said as the plane was coming into land at Bristol it did an "emergency take-off mid air" after being told of a "security threat" on the ground.
"It was pretty scary... we were [later] told we were only 1,500m from the ground and they [the pilot] had to override the automatic descent after being told to abort the landing."
Police are to interview every passenger on board a plane from Alicante, after a "bomb threat" note was found, prompting the evacuation of Bristol Airport.