According to BBC Sport journalist Simon Stone, the 22-year-old will join the Saints on a permanent deal.
The England Under-21 international spent last season out on loan at Championship outfit Norwich City, with his impressive performances doing enough to catch the eye of senior officials at Southampton.
Breaking the news on Twitter, Stone wrote: "Angus Gunn close to joining @SouthamptonFC from Man City. Would be permanent deal. Other clubs interested but Saints head the queue. Very highly-rated keeper.
Giving his further input and analysis of the situation, Stone added: "This is one of those tough situations. Everyone at City - and many other places - know this lad is going to be very good and potential England keeper one day.
This is one of those tough situations. Everyone at City - and many other places - know this lad is going to be very good and potential England keeper one day. But Ederson number one. Bravo on big money and Grimshaw just signed new deal. He can go somewhere good and play.— Simon Stone (@sistoney67) July 9, 2018
"But Ederson number one," he added, "Bravo on big money and Grimshaw just signed new deal. He can go somewhere good and play."
Gunn sparked further speculation on Twitter, when he responded jokingly to a tweet from Southampton midfielder Harrison Reed.
However, it's worth noting that the duo played together for Norwich last season, so not too much can be read into that one.
Gunn looks set to become Southampton's third signing of the summer, following the arrivals of Celtic midfielder Stuart Armstrong and winger Mohamed Elyounoussi, from FC Basel.
Southampton seem keen to strengthen their defensive unit, with Denmark defender Jannik Vestergaard set to have a medical on the south coast on Monday.