Eriksson has been in the headlines this week after revelations about his alleged relationship with fellow Swede and television personality Ulrika Jonsson.
A spokeswoman for BBC books said: "We have made an offer for the authorised biography. It is due for publication next year.
"It is going to be an up-to-date biography. It is still very much in the preparation stages."
An FA spokesman said: "We are fully aware of this publication. It was an agreement that Sven had made prior to taking his position as England coach.
"Therefore, the detail of the book will be with regard to his life and career up to that point.
"Sven does have right of approval and clearly if there was to be any discussion in the publication referring to his role with England, or during the World Cup, he would have an element of control over that."
Eriksson's private life hit the headlines last week after newspapers said he was involved with television presenter Jonsson.
Jonsson confirmed on Wednesday that she had a relationship with Eriksson and is also writing her biography.