The 21-year-old's future has been the subject of much speculation in recent weeks, with Man City said to have identified him as their leading target for the summer transfer window.
According to BBC Sport, an agreement with Leipzig has now been reached, with the treble winners set to pay €90m (£77m) for the Croatia international.
The report claims that there will not be any add-ons in the deal, meaning that Manchester United's £80m swoop for Harry Maguire from Leicester City in 2019 remains the British record for a defender.
Gvardiol will allegedly travel to Manchester later this week for a medical, meaning that he could potentially be available for the Community Shield against Arsenal on Sunday.
The centre-back made the move to Leipzig from Dinamo Zagreb in 2020, and he has represented the German side on 88 occasions in all competitions, including 59 appearances in the Bundesliga.
Gvardiol has been capped on 21 occasions by Croatia and played a key role in his national team reaching the semi-finals of the 2022 World Cup in Qatar.
The defender was also on the scoresheet in the third-place play-off against Morocco at the tournament, which Croatia won 2-1 to finish third behind Argentina and France.
Man City are already well-stocked in terms of central defenders, with John Stones, Manuel Akanji, Ruben Dias and Nathan Ake all important players for the club.
However, Kyle Walker and Aymeric Laporte are both being linked with moves away from the club, while Joao Cancelo is also set to leave, which would open up further space in the squad to make a new addition.
Man City have been relatively quiet this summer in terms of incomings, with Mateo Kovacic proving to be their only signing, making the move from Chelsea.
There have been four departures, though, with Benjamin Mendy and Ilkay Gundogan leaving on free transfers, while Yangel Herrera and Riyad Mahrez have joined Girona and Al-Ahli respectively.
Gvardiol's Premier League debut for Pep Guardiola's side could potentially come against Burnley on August 11.
Man City concluded their pre-season with a 2-1 defeat to Atletico Madrid on Sunday, and they are now building towards their clash with Arsenal in the Community Shield.