The British filmmaker directed more than 40 films including Shirley Valentine and Educating Rita.
He also worked with Roger Moore and Sean Connery on three 007 films You Only Live Twice, The Spy Who Loved Me and Moonraker.
He is thought to have passed away at his home in Monaco on February 23.
Mr Gilbert will be “sorely missed”, Bond producers Michael G. Wilson and Barbara Broccoli said.
They added: “It is with great sadness that we learn of the passing of our dear friend Lewis Gilbert.
“Lewis was a true gentleman. He made an enormous contribution to the British film industry as well as the Bond films, directing You Only Live Twice, The Spy Who Loved Me and Moonraker.
“His films are not only loved by us but are considered classics within the series.”
On Sir Roger Moore’s official Twitter, which is managed by his family, there was a tribute to the director that read: “Lewis Gilbert … was one of Sir Roger’s favourite directors.
“The industry and Bond world will miss him although his legacy lives on forever. RIP Lewis.”
Peyton Reed, director of Ant-Man and the Wasp, meanwhile wrote on Twitter: “SPY was the first Bond film I saw in the theater. (And I have a tiny homage to MOONRAKER in ANT-MAN AND THE WASP.) Rest in Peace.”
Comedian David Walliams also paid tribute to the director, writing: “The rooftop fight from the classic Bond @007 film ‘You Only Live Twice’ is one of my favourites scenes from the series & was directed by the late great #LewisGilbert.”