Kohli, a national hero and the most famous cricketer on the planet, had been named in the squad for Thursday's series opener in Hyderabad but requested a leave of absence.
The Board of Control for Cricket in India did not offer details about the 35-year-old's circumstances but stated: "certain personal situations demand his presence and undivided attention".
The news follows England batter Harry Brook flying home to be with his family, with both sides losing a key member of their batting unit.
Kohli also missed two T20s against Afghanistan earlier this month for personal reasons, but came back to the side to complete the series.
A statement from the BCCI secretary Jay Shah read: "Mr Virat Kohli has requested to be withdrawn from the first two Tests of the upcoming IDFC First Bank Test series against England, citing personal reasons.
"Virat has spoken to captain Rohit Sharma, the team management and the selectors and has emphasised that while representing the country has always been his top priority, certain personal situations demand his presence and undivided attention.
"The BCCI respects his decision and the Board and team management has extended its support to the star batter and is confident in the abilities of the remaining squad members to step up and deliver commendable performances in the Test series.
"The BCCI requests the media and fans to respect Virat Kohli's privacy during this time and refrain from speculating on the nature of his personal reasons. The focus should remain on supporting the Indian cricket team as they embark on the upcoming challenges in the Test series."
Kohli boasts nearly 9,000 Test runs and 29 centuries, leaving a major hole in the India XI. He has also excelled on home soil against England, averaging 56.38 in 13 Tests.
India have named just six specialist batters in their squad, as well as wicketkeepers KS Bharat and Dhruv Jurel, with a replacement for Kohli set to be announced.
India are already without one of their first-choice seamers, with Mohammed Shami still working his way back to fitness.
