"Aron Modig is conscious and staff from the Swedish embassy are on their way to the hospital," confirmed KDU's press secretary Kalle Bäck in a statement on Monday.
The dead have been identified as Oswaldo Paya, one of Cuba's most well-known dissidents, and opposition politician Harold Capero.
A fourth passenger, Spanish citizen Angel Carromero Barrios, was injured in the crash which occurred in the Granma province some 750 kilometres east of Havanna.
The car drove off the road between the provincial capital Bayamo and Las Tunas of at 1.50pm local time on Sunday. The car then collided with a tree.
According to a CNN report, Oswaldo Paya's daughter has forwarded claims that a further car was involved in the crash and that the vehicle deliberately drove into the car carrying Aron Modig.
This information was however denied by the Cuban government in a statement on Sunday, citing eye witnesses.
Modig was reported to have visited Cuba to offer his support for the pro-democracy movement.
Oswaldo Paya received the Sakharov Prize in 2002 and received a nomination for the Nobel Peace Prize in 2005 by the former Czech president Václav Havel.
He was 60-years-old when he died.