"He has passed away," a family member told PTI, without disclosing any further details.
The 64-year-old actor was in Dubai apparently for a concert when he suffered the heart attack. His body will be brought to Mumbai later in the day after completing the formalities in Dubai. (In Pics: Bollywood Remembers Farooq Sheikh)
Mr Sheikh was best known for his roles in Satyajit Ray's Shatranj Ke Khiladi, Sai Paranjpe's Chashme Baddoor and Sagar Sarhadi's Bazaar. He started his career in 1973 with M S Sathyu's acclaimed film on India's partition, Garm Hawa.
He won the 2010 National Film Award for Best Supporting Actor for Sanjay Puran Singh Chauhan's Lahore.
The actor was last seen in Club 60, released earlier this month. His recent films include Shanghai and Yeh Jawani Hai Deewani.
"T 1336 -God !! Farooq Sheikh passes away !!? A true gentleman, a wonderful colleague ! A quiet honesty about him. Very very sad !" Amitabh Bachchan tweeted.
"He was one of the finest actors we had," an emotional Deepti Naval, who worked with Farooq Sheikh in some of his most successful films, told NDTV.
Writer and activist Javed Akhtar remembered him as an avid reader "who used to recommend books every time we met."
"#RIPFarooqShaikh The last time we met, we talked politics. I wish we has talked film and acting. Its what we love most. I would have learned," filmmaker Shekhar Kapur tweeted.
Filmmaker Mahesh Bhatt posted, "Alvida Farooq bhai the warmth of your smile lingers in our memory!"
"Farooq Sheikh passes away! Shocked, numbed and saddened. Was supposed to be shooting with him today. A gentleman actor is no more!" actor Boman Irani tweeted.
"RIP Farooq Sheikh. Immensely talented, extremely courteous. A thorough gentleman," actress-turned-politician Smriti Irani posted on Twitter.