The club have exchanged definitive agreements with GMR as part of a "significant strategic partnership", making them the first county to be owned by an overseas franchise.
Under the deal, ratified by the England and Wales Cricket Board (ECB), the Indian conglomerate acquires a majority stake and is expected to complete a 100 percent takeover within two years.
Rod Bransgrove, chairman of Hampshire Sport and Leisure Holdings, said in a statement on Monday that the deal was the "fulfilment of a dream".
"Becoming the first English cricket club to join an international cricket group will open exciting new opportunities as we embrace the globalisation of this great sport," he said.
GMR owns a 50% stake in Delhi Capitals, Dubai Capitals and Pretoria Capitals, as well as investing in US Major League Cricket team the Seattle Orcas.
Kiran Kumar Grandhi, corporate chairman of GMR Group, said: "Our vision is to transform sports into a platform that unites people and cultures, drives global excellence, and nurtures the creation of future world champions."
ECB chief executive Richard Gould said the announcement "demonstrates the global interest in investing in cricket in England and Wales, and underlines the continuing appeal of county cricket".
Hampshire finished second in division one of the just-concluded County Championship.
