The 44-year-old is currently working as a TV pundit after leaving his position as assistant to Louis van Gaal at United when the Dutchman departed Old Trafford at the end of the 2016 season.
Giggs, who played for Wales 64 times, has made no secret of his desire to succeed Chris Coleman as national team manager, which last year he described as the ideal job.
Coleman took Wales to the semi-finals of EURO 2016 but stepped down last November after they failed to qualify for the 2018 FIFA World Cup. The 47-year-old then took over as manager of Sunderland in the Championship.
Giggs is reported to be one of a number of candidates being considered for the vacancy by the seven-man Football Association of Wales (FAW) committee, led by chief-executive Jonathan Ford.
An appointment is expected by the end of January. A FAW press spokesman did not immediately respond to a Reuters request for comment.
"I know the players and the system. We've got a world-class player in Gareth Bale, some great players just beneath, some really good young players coming through like Ethan Ampadu and Ben Woodburn," said Giggs in an interview with the Times last month.
"It would be a great job."
Although Giggs's only managerial experience was as Manchester United caretaker in succession to David Moyes, Mark Hughes was similarly inexperienced when he was appointed in 1999 and Wales have often appointed former players at the start of their managerial careers.
Giggs said it would have been a mistake to go into management immediately after playing. "I wasn't ready when I had those four games in charge at United, but I'm a lot better qualified to manage today after those two years with Louis. That was a fantastic experience," he said.
Hughes, who was dismissed by Premier League side Stoke City earlier this month, is one of Giggs's main rivals for the position.