The 23-year-old will cost an initial £12.5m, with up to £1.5m further due in performance-related add-ons. Petrovic will now travel to complete his medical in the coming days.
Petrovic will sign a seven-year deal with an option for a further year to replace Kepa Arrizabalaga, who last week joined Real Madrid on a season-long loan.
The two-time Serbia international will initially deputise for Chelsea’s no.1 goalkeeper Robert Sanchez but the Blues believe him capable of becoming first choice in the future.
Former Chelsea goalkeeper Kevin Hitchcock served as his goalkeeping coach in MLS.
Chelsea have almost completed their summer spending, with Mauricio Pochettino expected to add one more forward before the transfer window closes at 11pm on September 1.