The 85-year-old actor and wrestler was hospitalised on July 7 in a critical condition.
Shah Rukh Khan told followers on Twitter: "Dara Singh was our very own superman. Wrestlers r made of sweat,determination & a hard to find alloy called guts...most apt for Dara Singhji our very own Superman.Will miss u sir."
Amitabh Bachchan tweeted earlier: "Dara Singh ji passed away this morning. A great Indian and one of the finest humans..An entire era of his celebrated presence gone!"
Singh was last seen playing grandfather to Kareena Kapoor's character in the romance Jab We Met in 2007.
The actor made the transition from wrestling in 1952 after starring in a series of action roles in films such as Tarzan Comes to Delhi, Samson, King Kong and Hercules.
He also had a long-running role in the Indian television epic Ramayan as the monkey god Hanuman.
It was reported that he was visited in hospital by celebrity friends Raza Murad, Farah Khan and Dharmendra.
A source told The Daily Bhaskar that Salman Khan ordered a minute's silence on the set of Ek Tha Tiger to mark the legend's passing away.