The Board of Control for Cricket in India (BCCI) has confirmed that the batsman suffered the injury while playing in the Indian Premier League.
"[Kohli] will now undergo a period of rehabilitation under the supervision of the BCCI medical team," BBC Sport quotes the governing body as saying.
"He will begin training and subsequently undergo a fitness test in Bengaluru from June 15. The BCCI medical team is confident that Mr Kohli will regain full fitness ahead of India's upcoming tours to Ireland and England."
Kohli was scheduled to play at least six games for Surrey this summer after signing his first overseas contract with the team in May.