He was 65.
"He has passed away," a family member told PTI, without disclosing any further details.
The actor was in Dubai apparently for a concert when he suffered the heart attack on late Friday night.
His body will be brought to Mumbai later in the day after completing the formalities in Dubai.
Sheikh made a tremendous contribution in theatre, films and TV as well. He started his career in Bollywood in 1973 with 'Garm Hava' and is best remembered for his performances in films like 'Shatranj Ke Khiladi', 'Chashme Buddoor', 'Kissi Se Na Kehna', 'Noorie'.
He also hosted the popular Zee TV show "Jeena issi ka naam hai" in which he interviewed many Bollywood celebrities.