Ferguson led United for 26 years before retiring this summer and being replaced by David Moyes.
He previously announced his intention to stand down in 2002 and although Ferguson later reversed that decision, Eriksson claims he was approached to replace the Scot and accepted a deal.
In an extract from the book reproduced in the Mail on Sunday, Eriksson wrote: "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. A contract was signed - I was United's new manager."
The 2006 tournament itself gave rise to further issues as Eriksson selected Rooney, who had been battling a broken metatarsal in the weeks leading up to the squad announcement.
Ferguson, Eriksson claims, had said Rooney would not be available, giving rise to a difference of opinions between medical staff with club and country - with England doctor Leif Sward ultimately winning out.
Eriksson wrote: "Leif and I met Ferguson and United's doctor at the United training ground. 'He cannot play in the World Cup', Ferguson said flatly. The doctor brought out some X-rays that he said showed Rooney's broken bone would not heal in time.
"When the doctor finished, Leif looked him in the eye. 'Why do you sit here and lie to me?' Leif asked.
"He was one of Europe's foremost specialists on this kind of injury. I just wish I could have filmed Ferguson's face when Leif explained that Wayne's break would heal in time for the World Cup.
"When Leif had finished, I turned to Ferguson. 'Sorry, Alex,' I said. 'I will pick Rooney'."
Rooney went to the tournament but did not score a single goal and was sent off for stamping on Ricardo Carvalho in England's quarter-final against Portugal, which they lost on penalties.