The Swede, currently in charge at Chinese Super League outfit Guangzhou R&F, makes the assertion in his new book Sven: My Story, which is being serialised by the Mail on Sunday.
Having led England to the quarter-finals of the 2002 FIFA World Cup in Japan and South Korea, Eriksson also went on to reach the last eight at both UEFA Euro 2004 and the 2006 World Cup in Germany.
However, the 65-year-old has revealed he had the chance to make a return to club football much earlier than 2007, when he became manager of United's rivals Manchester City.
"That (becoming United boss) is what very nearly happened," writes Eriksson. "Ferguson had announced that the 2001/02 season would be his last. He had managed the team for 16 years and won everything there was to win, including the Premier League seven times.
"One day I got a phone call from Pini Zahavi, a leading football agent. He wanted to know if I could come for breakfast at a club in London the following morning.
"He did not want to say on the phone what it was about. It was very secretive. ‘Sure,’ I said, ‘no problem.’
"When I arrived, Pini was there with Peter Kenyon, the chief executive of Manchester United. Straight off the bat, Kenyon asked me did I want the job as manager of Manchester United as of next season? I didn’t think about it.
"‘Yes,’ I said. ‘I do.’
"I knew it would be tricky. I had a contract with England until the 2006 World Cup and I would be severely criticised if I broke that contract.
"But this was an opportunity to manage Manchester United, probably the biggest club in the world. I would be able to stay with England through the World Cup. My appointment would not be made official until after the tournament.
"A contract was signed – I was United’s new manager."
The alleged deal hit a snag when Ferguson made a U-turn on his retirement plans, opting to remain in the hot seat at Old Trafford.
"A couple of weeks passed and Pini called again," Eriksson continues. "He wanted another meeting. When I got there, I knew something was wrong.
"Kenyon explained that Ferguson had changed his mind. He did not want to leave the club after all, but had agreed to stay in the job for another three years."
Ferguson went on to win another six Premier League titles and a second UEFA Champions League trophy as United manager before retiring at the end of last season to be replaced by David Moyes.
Eriksson, meanwhile, has held a string of positions since leaving England, including national posts with both Mexico and the Ivory Coast.
