Giggs, 44, has been the clear favourite for the Wales job since he declared his interest last month, saying: “I’ve played for Wales and I’ve said that I want to go back into coaching.
“Obviously that is one of the top jobs.”
Giggs was interviewed for the Wales job last week along with former international teammate Craig Bellamy and Osian Roberts, Coleman’s former assistant who is also the FAW’s technical director.
Former Wales defender Mark Bowen was also interviewed after leaving his role as Stoke’s assistant manager a few days earlier.
Giggs’s contract, which will reportedly take him up to the 2022 World Cup, was tied up over the weekend.
The FAW was keen to make the appointment before the Uefa Nations League draw, which takes place in Switzerland on January 24.
Giggs, who won 64 Wales caps between 1991 and 2007, has been out of football for 18 months since leaving the coaching staff at Manchester United.
He spent two seasons as Louis van Gaal’s assistant coach, but he left Old Trafford in the summer of 2016 following Jose Mourinho’s appointment as manager.
That ended a long association with the club where he made a record 963 appearances as a player, scoring 168 goals.
Coleman spent nearly six years as Wales manager before leaving to take over Championship strugglers Sunderland in November.
He became the most successful manager in Welsh football history when he guided the country to the semifinals of Euro 2016 – Wales’s first major tournament for 58 years. But they failed to qualify for this year’s World Cup in Russia.