Claude Puel was fired as Saints boss on Wednesday, despite guiding them to an eighth-placed finish last term.
Giggs was linked with the Swansea job after leaving as United assistant boss last year before Jose Mourinho succeeded Louis van Gaal.
And now it seems he has set his sights on taking over Saints.
The Daily Mail claim the Welshman wants to take his first proper steps into management and see the south Coast outfit as the ideal place to do so.
Former Borussia Dortmund boss Thomas Tuchel had been considered, but the Mail claim the role would interest Giggs.
Frank de Boer and former Alaves manager Mauricio Pellegrino are also on the shortlist.
Giggs admitted last month he struggled to cope with leaving United, as he said: “I made the decision to see a psychiatrist to learn how best to cope and some of the suggestions he made served me well in adapting to a new life outside of United.
“He suggested that I keep busy in the immediate aftermath, and I did that going to the European Championship in France last summer as a pundit and then to India for a futsal tournament I had been invited to play in.
“There were little things too. I joined a gym for the first time in my life, and his simple suggestion that I join one half an hour from my home forced me to make a routine.
“My whole life had been mapped out during my 28 years at United. I was at a major point of transition in my professional and private life. I knew it would be difficult psychologically.”
