Gvardiol, 21, will next undergo a medical and is set to travel to England imminently.
He is unlikely to make his debut in Sunday's Community Shield clash with Arsenal at Wembley but should be signed in time to be available for City's league opener against Burnley at Turf Moor on Aug. 11.
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 Manchester United paid Leicester City for Harry Maguire in 2019.
Gvardiol will become Pep Guardiola's second summer signing after the arrival of Mateo Kovacic from Chelsea.