The Addicks are understood to be close to agreeing a deal with the 42-year-old and a compensation package with Swindon, although the South London club insisted on Thursday morning that they remain in talks with other potential candidates.
Garner came through the coaching ranks at Crystal Palace and worked under the likes of Ian Holloway, Tony Pulis, Neil Warnock and Alan Pardew at Selhurst Park.
He has previously managed Bristol Rovers and led Swindon to the League Two play-off semi-finals this season, where they lost on penalties to Port Vale.
Charlton owner Thomas Sandgaard sacked manager Johnnie Jackson at the end of the season and has targeted a successor who can lead the club to promotion from League One with a modern, high-pressing style. Jackson has since taken over at League Two side AFC Wimbledon.