Kovacic, who won the Champions League three times with Real and once with Chelsea, could be one of several departures from the London club as new manager Mauricio Pochettino trims his squad.
The club confirmed the signing via their Twitter account on Tuesday.
The move marks a return to Stamford Bridge for Frank who enjoyed an illustrious playing career with the Blues.
For the second time in three days, Manchester City have defeated Chelsea twice, handing The Blues a 4-0 thrashing at the Etihad Stadium on Sunday evening.
It was reported in British media that Mr Willoughby had been sending the messages to a football finance agent Catalina Kim from last year.
Graham Potter said he is incredibly proud and happy to represent Chelsea FC
Discussions took place between the Chelsea hierarchy and Potter’s representatives into Thursday morning.
Chelsea Football Club have today parted ways with head coach Thomas Tuchel.
UEFA announced that the finances of Man City, Barcelona, Chelsea and the other 15 clubs will be monitored closely for the remainder of the year.
Of the 200 people listed as the richest Russians by Forbes magazine, barely a quarter are on Western sanctions lists, according to Mr Navalny.
