The Daily Mail says there is growing expectation at Craven Cottage that Parker will leave the club in the coming days.
That would pave the way for Bournemouth to make their move to appoint Parker, who is their No 1 managerial target.
The situation is expected to accelerate in the next few days.
Sources claim a deal for Parker to join the Cherries would be a formality, though the club are waiting on his exit from Fulham before formalising their interest.
Jonathan Woodgate has yet to formally leave his role as head coach at Bournemouth but had only signed a deal until the end of the season to become the club's interim head coach.