Leipzig had initially valued Gvardiol at €100m, a stance that stalled talks earlier this summer.
Leipzig's valuation would have made Gvardiol the most expensive defender in the world, but City's negotiated fee is less than the £80m ($102m) Manchester United paid Leicester City for Harry Maguire in 2019.
The 21-year-old joined Leipzig for a fee of around €19m in 2020 and, after spending a season back on loan at Dinamo Zagreb, established himself as a key figure for a team that won back-to-back DFB-Pokals.
He has made 21 appearances for Croatia and was part of the team that finished third at the 2022 World Cup in Qatar.
Since Pep Guardiola's side lifted the treble last season, Ilkay Gündogan and Riyad Mahrez have both left, and there could yet be more outgoings, with interest in Bernardo Silva, Kyle Walker, Kalvin Phillips and Aymeric Laporte.
Information from ESPN's Rob Dawson contributed to this report.