The news comes weeks after Oscar was moved into the care of the Children's Hospice following his two-year battle against the disease.
At that time, the five-year-old's suffering was so great that he told his parents: "I don't want to be a boy anymore."
In the latest update to an online diary documenting Oscar's fight, Leona wrote: "In the past four weeks Oscar has suffered more pain than we could have imagined, and more anguish than any five-year-old should ever know."
She said that she and her husband Stephen had been "pushed to the limit of what any parent could tolerate in terms of watching their child suffer".
The family have asked for privacy in the coming days.
Oscar's funeral, which the family have said will be a "celebration" of his life, will be held in St Bernard's Church, Glengormley, on Sunday.
In a statement, the family said: "We welcome friends, family and supporters of Oscar to join us in procession from the NI Children's Hospice at 10.15am on May 11 and onwards to St Bernard's Church."
A strictly private cremation will follow.
An appeal by Oscar's family raised hundreds of thousands of pounds over the last three years. The campaign won many high-profile friends and memorably Oscar wreaked havoc during a visit to the offices of Northern Ireland's First Minister Peter Robinson and Deputy First Minister Martin McGuinness at Stormont.
First Minister Peter Robinson said: "I extend my heartfelt sympathies to Leona, Stephen, little Isobella and the entire Knox family on the death of Oscar. My thoughts and prayers are with them all at this profoundly sad time.
“Their sense of loss and devastation is immeasurable. When I met Oscar I was struck by his infinite energy and sense of fun. He had a tremendous capacity to overcome the challenges he and his family faced together.
Deputy First Minister Martin McGuinness said: "Oscar touched the hearts and minds of everyone and inspired countless people through his positive attitude and resilience.
"The day Oscar and his family visited Stormont was one of my most uplifting days in politics and you couldn’t help but be enthused by his vibrancy and spirit."
In a statement today, Celtic Football Club said: "This is absolutely devastating news and our thoughts and prayers are with Oscar's parents, Stephen and Leona, and his little sister, Izzie, at this desperately sad time.
"We can't even begin to understand the pain of their loss, but I hope that there will be a small measure of consolation in knowing that there was genuine love and affection for Oscar from the Celtic Family throughout the world.
"Oscar's courage throughout his illness was truly inspirational, and it was a real pleasure for everyone at the club to meet Oscar when he was our team mascot last July."
Former world champion boxer Barry McGuigan tweeted: "Very sad news to hear of the passing of @Wee_Oscar RIP he fought to the bitter end, what an amazing young boy. Deepest sympathies to family.""
Little Oscar has touched the hearts of people with his brave fight since being diagnosed with high-risk neuroblastoma – a rare and aggressive childhood cancer – in November 2011.
His parents went public in a bid to raise awareness and funds for Oscar's care.
A huge support campaign was launched, backed by celebrities and sports stars including Belfast Olympic medal-winning boxers Paddy Barnes and Michael Conlan.
Since early 2012 Stephen and Leona have kept an online diary of Oscar's fight.
In 2011 Oscar Knox (5), from Mallusk, was diagnosed with high-risk neuroblastoma, which affects one in every 100,000 children. He also has the chromosome disorder Jacobsen syndrome, which is thought to affect only 250 children worldwide.
Oscar has received more than 30 blood transfusions and almost 140 platelet transfusions in the past 26 months. Oscar's parents Stephen and Leona run a blog and fundraising campaign, The Wee Oscar Knox Appeal. Around 140,000 people follow Oscar on Facebook and Twitter @Wee_Oscar