The singer was originally diagnosed last year but has since been informed by doctors that he is in remission following chemotherapy and radiotherapy.
Gibb, who recently received a standing ovation at the London Palladium, now intends to fulfil his work commitments and has planned a trip to America to appear on Piers Morgan's CNN show on April 17, the Sunday Mirror reports.
"I could not have done it without the love of [wife] Dwina. She has never left my side and is a wonderful person. I love her so much for what she has done for me. I am beating cancer and I can't wait to carry on with my work," Gibb told a close friend.
"My work means everything to me and keeps me going. I want everyone, all my fans, to know I am okay. Their love helps me so much."
Gibb has been working with son Robin-John on The Titanic Requiem, a classical music project marking the 100th anniversary of the sinking of the Titanic. He is due to perform with the Royal Philharmonic Orchestra at the work's premiere at London's County Hall on April 10.
A source added: "There will be difficult times ahead but Robin's remarkable positivity along with his response to the treatment is very encouraging and has thrilled everyone. They were the dark days up until Christmas and you could have never seen where we are now.
"Every day was bad news, whereas now, we are getting good news. Getting on a plane to anywhere, let alone New York, would not have happened a few months ago. Doctors wouldn't let him. His focus has been to get back in shape and this project has driven him on."
Gibb revealed last month how the disease had nearly gone and plans for a Bee Gees comeback were subsequently drawn up.
> Bee Gees star Robin Gibb back in hospital for cancer treatment
Watch the video to the Bee Gees' 1977 hit single 'How Deep Is Your Love' below: