"I didn't see David Cameron. I didn't know it was David Cameron until they let me out of the police van an hour later," said Dean Farley, who was eventually released without charge.
"It begs the question -- how good is Cameron's security if I managed to run between it before they stopped me?" he was quoted as saying.
Farley collided with Cameron on Monday as he left the city's Civic Hall in Leeds and was driven away from the scene, apparently uninjured.
A member of his security team intervened and bundled Farley away as the Prime Minister got into a waiting vehicle. He was arrested and later released.
Farley, who was carrying a towel but no ID at the time, said it had been "harrowing" to find himself "harangued and manhandled" by police and not being told why he had been arrested.
He insisted he was "not particularly political" and was just going out on his daily lunchtime jog to the gym when he ran into a "bunch of men in suits".
Cameron has downplayed the incident, now the subject of a police review.
West Yorkshire Police said "nothing sinister" had taken place but the Metropolitan Police, which provides personal security for the prime minister, said there would be a review of the incident.
The prime minister was in Leeds to launch government plans to upgrade rail links in the north of England.
"I'm quite shook up. I was almost in shock, like I'd been in an accident or something," Farley said.
The media reaction to the incident had been "insane", he said, adding that many of his friends wanted to buy him a drink and he could see the "funny side".
"I kind of wish I had been protesting something or I had had something to say", Farley said.
The Prime Minister's close security is generally provided by officers from SO1 Specialist Protection, part of the Metropolitan Police's Protection Command.
Labour MP Keith Vaz said he would be "astonished" if there was not a review of procedures.

"It could have ended in a completely different scenario," he said.

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