Kohli should be available for three County Championship games against Hampshire, Somerset and Yorkshire.
The right-handed batsman could also feature in three one-day Cup matches against Kent, Middlesex and Glamorgan.
"It has long been an ambition of mine to play county cricket and I am thankful to Alec Stewart and Surrey for allowing me the opportunity to join them during their 2018 season," said Kohli in a Surrey statement.
India take on England in three-match T20 and one-day series in July before a five-match test series.
"We are thrilled to have signed the biggest name in world cricket for the month of June," said Surrey director of cricket Alec Stewart.
"Playing and training alongside Virat will be a massive benefit for our players who will have the opportunity to learn so much from him.
"At a time when there is much discussion around the future of county cricket, the arrival of Virat should give our domestic game a massive boost and positive exposure around the cricketing world which in turn can benefit every county."