The 49-year-old former Australia boss, who won the A-League with the Central Coast Mariners last season, told the Australian that the Blades had interviewed him for their vacant position.
But, after a video call which included United director and former manager Dave Bassett, Arnold said he had no hesitation in turning the job down.
He told the newspaper: "In all honesty, I didn't have to think too long or too hard about it. It just wasn't for me.
"Forget that the financial offer in itself was quite poor, but I would have been taking three steps backwards in my development as a professional coach if I had gone.
"I walked out of that meeting with the club's board and football director Dave Bassett, and said 'Wow'. It just blew me away.
"Being one of the bigger lower league clubs, I was expecting a lot more in terms of how they wanted to develop as a club on the field.
"But they just didn't seem interested when I was talking about playing a short passing game and taking the football another level up. It was about smashing the ball long and working on set-pieces... it was prehistoric stuff.
"That's not the way to develop a football team."
United, who were play-off semi-final losers to Yeovil, sacked Danny Wilson with five games of last season to go and have been searching for a full-time replacement ever since.
Sheffield United were not available to comment when contacted on Sunday by the media.