Judges of the annual Dave award presented Walsh with the prize for his gag: "I just deleted all the German names off my phone. It's Hans free."
But now, stand-up Pete Cunningham has claimed that he first coined the joke "seven or eight years ago" and shared it with Walsh during a Twitter exchange.
Cunningham, who performs as alter-ego Frank Sanazi, told BBC Scotland that he is on friendly terms with Walsh and they often "go backwards and forwards with comments and puns about each other's stuff".
"Frank Sanazi is my most popular character and has a bit of a cult following," he said. "I was doing a gag seven or eight years ago - some new material. One of the gags is that I pretend that I have an agent called Harvey Goldberg, because being Frank Sanazi - you see what I mean, having a Jewish agent - that's the joke.
"I say my agent likes me to take the German names off my phone to make it Hans free. Darren has used the punchline for that joke."
@DarrenWalshPuns I've removed all ze German names off my phone it's now Hans Free !— Frank Sanazi (@FrankSanazi) April 14, 2015
TV channel Dave has defended Walsh and said it is "entirely confident" that he is the rightful winner.
The jokes for Dave's Joke of the Fringe are submitted by 10 extremely experienced comedy critics from shows performed at the Edinburgh Fringe," a spokesperson said. "Each submission is checked and cross-referenced for originality before being put to a public vote."
Cunningham, who also has a Fringe show, added that he "wasn't bitter" about Walsh's success as comics often say "there are only two or three jokes in the world and they are all sort of spun around".
However, he did have one parting shot. "It's quite ironic that the best gag at the Fringe is one of the weakest gags in my show," he said. Ouch.
10 best Edinburgh jokes
1: Darren Walsh: “I just deleted all the German names off my phone. It’s Hans-free.”
2: Stewart Francis: “Kim Kardashian is saddled with a huge arse... but enough about Kanye West.”
3: Adam Hess: “Surely every car is a people carrier?”
4: Masai Graham: “What’s the difference between a ‘hippo’ and a ‘Zippo’? One is really heavy, the other is a little lighter.”
5: Dave Green: “If I could take just one thing to a desert island I probably wouldn’t go.”
6: Mark Nelson: “Jesus fed 5,000 people with two fishes and a loaf of bread. That’s not a miracle. That’s tapas.”
7: Tom Parry: “Red sky at night. Shepherd’s delight. Blue sky at night. Day.”
=8: Alun Cochrane: “The first time I met my wife, I knew she was a keeper. She was wearing massive gloves.”
=8: Simon Munnery: “Clowns divorce. Custardy battle.”
10: Grace The Child: “They’re always telling me to live my dreams. But I don’t want to be naked in an exam I haven’t revised for...”Reuse content