The 23-year-old flew from Ireland to London to catch a connecting to Japan for the Games, which start on August 24.
However, he posted on Twitter that: “When the plane landed I was given my wheelchair back, except it was completely broken.”
In his tweet on Saturday explaining the situation, Flanagan included pictures which appeared to show the broken wheels.
“Ultimately the wheels have been damaged so much that they no longer spin, along with a lot of other dents and scrapes,” he wrote.
He explained that the staff at Heathrow were “extremely apologetic.”