Wales are on the lookout for a new boss following Coleman's resignation last month, and former international Giggs has thrown his name into the ring.
Giggs had a four-match spell as interim boss at United in 2014 following David Moyes' sacking, while he also worked as an assistant to Louis van Gaal until 2016.
The 44-year-old – capped 64 times by the national team – has failed to land a permanent managerial role despite being linked with numerous jobs but he is attracted to the Wales vacancy.
"Of course I'm interested," Giggs told Sky Sports News.
"I played for Wales, I've said that I want to go back into coaching and that is obviously one of the top jobs.
"I played for United, I played for Wales, those are the two jobs.
"I've not spoken to anyone at the moment but I'd definitely be interested."
Wales qualified for Euro 2016, where they reached the semi-finals, having previously failed to reach a major tournament since 1958.
However, Wales were unable to book their spot at the 2018 World Cup in Russia as Coleman left for Championship outfit Sunderland.