Attacking midfielder McAtee, 19, was named the 2021-22 Premier League 2 Player of the Season after enjoying a successful campaign with City's Under-23s side, recording 18 goals and seven assists in 22 league games.
Eighteen-year-old defensive midfielder Lavia, meanwhile, made 27 youth appearances across all competitions last season and, like McAtee, is highly regarded by senior boss Pep Guardiola.
According to the Daily Mail, via the Southern Daily Echo, Southampton are set to rival Leeds in the race for both McAtee and Lavia this summer.
Both clubs may face competition from Brighton & Hove Albion and Huddersfield Town for the signature of McAtee, with the latter understood to have scheduled talks with Man City over a proposed loan move.
The report adds that the Citizens could potentially use either McAtee or Lavia as a makeweight in their pursuit of Leeds midfielder Kalvin Phillips, who has been heavily linked with a move to the Etihad Stadium.
Jesse Marsch has already bolstered his squad this summer with three new arrivals, including midfield duo Brenden Aaronson and Marc Roca, while Southampton boss Ralph Hasenhuttl have already recruited one player from Man City, with goalkeeper Gavin Bazunu joining the club on a five-year contract.