Talks on the proposal have been held between Moscow and Abu Dhabi state investment fund Mubadala, The National newspaper reported citing Russian Industry Minister Denis Manturov.
He said Cuba planned to convert a former military base into "a big airport hub for Latin America".
The project could also include redeveloping a seaport and constructing a railway line.
A Mubadala spokesman told the Abu Dhabi-based National that the company was "regularly reviewing a number of different investment opportunities with its Russian partners".
In December, US President Barack Obama and Cuba's Raul Castro announced their two nations would begin normalising ties, ending a half-century of enmity between the former Cold War foes.