Neville is in talks with the humble Northern Premier League Division One North club with Giggs also likely to be involved in plans to transform the minor league outfit, according to the Manchester Evening News.
"Discussions are at a preliminary stage," a spokesman for Neville told BBC Sport.
Salford City, who play in front of gates of around 100, could become a hub for talented local youngsters, who would remain in the area with a view to being groomed for a seamless move into the professional game.
Both Neville and Giggs have links with Salford as Giggs grew up there and Neville lived nearby while United's old training ground 'The Cliff' was in the district.
Salford were formed in 1940 and have always played in the lower reaches of the non-League pyramid.
Neville, 38, retired in 2011 after playing for United for almost 20 years, while Giggs, 40, is still in United's first team having made nearly 1,000 appearances for the club.
The two are also business partners and own a restaurant in Manchester and have a stake in a new hotel being built opposite United's Old Trafford stadium.
Earlier this week their old United colleague Beckham announced plans to move into club ownership with the launch of a new Major League Soccer club in Miami.