City are set to pay £25 million ($32m) up front for the Croatia international, with another £5m ($6m) due in add-ons.
Sources have told ESPN that Chelsea were initially hoping to agree a deal worth closer to £35m ($44m) but reduced their demands in order to reach an agreement before June 30. Outgoings completed before then will fall under the first full tax year and help Chelsea comply with UEFA's Financial Fair Play rules.
Kovacic will undergo a medical before joining Pep Guardiola's squad ahead of City's preseason tour of Japan and South Korea, which begins at the end of July.
Guardiola and director of football Txiki Begiristain are still hoping to convince Ilkay Gundogan to stay despite interest from Barcelona and Arsenal, while there are also doubts about Bernardo Silva's future at the Etihad.
Gundogan will officially become a free agent June 30 and has contract offers on the table from both City and Barca.
Silva is under contract until 2025 but is keen for a new challenge after six years in England. Sources have told ESPN he has interest from Barcelona and Paris Saint-Germain but that City will demand a significant fee before allowing him to leave.
City, according to sources, also hold an interest in West Ham's Declan Rice but are yet to table a bid. Arsenal have already had two offers for the England midfielder knocked back.
Sources have told ESPN that West Ham are keen to explore a deal for Rice with City that would see Kalvin Phillips move to the London Stadium.
Phillips struggled for opportunities in his first season at the Etihad but has said he wants to stay and fight for his place. West Ham boss David Moyes previously tried to sign Phillips during his time at Leeds United.