The former civil servant, 57, resigned in 2010 after just nine months in charge but is now thought to be considering a potential return.
Dyke announced on Thursday he would stand down as chairman in July over opposition to his governance reforms.
Watmore, whose son Duncan plays for Sunderland, was a member of the England 2015 Rugby World Cup board.
Dyke, appointed in 2013, had intended to stand for a further year.
However, he said that opposition to proposed reforms from some FA councillors and a minority of board members had made him reconsider.