Giggs eventually stood down as Wales boss in June of this year ahead of a domestic violence trial which is ongoing. He has pleaded not guilty to all charges.
After taking Wales to the last 16 of Euro 2020, Page – who previously served as Giggs’ assistant – led the country to their first World Cup since 1958, sealing qualification via a memorable play-off victory over Ukraine in Cardiff in June.
Williams says he and FAW chief executive Noel Mooney have held advanced talks with Page over fresh terms.
“Noel Mooney and myself are working with Rob and his agent on an exciting, long-term contract,” Williams told BBC Sport Wales.
“We’re dotting the i’s and crossing the t’s and fingers crossed we’ll be able to make an announcement in the forthcoming weeks.”
Wales begin their World Cup campaign against the United States on November 21 before also facing Iran and England in Group B.