The 60-year-old told listeners about his diagnosis while presenting The Folk Show on Wednesday evening.
“I’m sad to say that I’ve got some cancerous tongue and lymph node issues,” he said.
The presenter added he hoped to be back on air in the New Year, “or sooner, if I feel well enough”.
“It’s all been caught very early and so everything should be fine.”
Radcliffe confirmed he would broadcast his 6 Music show with Stuart Maconie on Thursday as usual, before going away.
He also shared the news on his Twitter account.
A BBC spokesman said: “Everyone at Radio 2 and 6 Music wishes Mark well and we look forward to welcoming him back when he’s ready.”
Ricky Ross will present the Radio 2 Folk Show for the next few weeks.
Fellow presenters on the station were quick to tweet their well wishes to Radcliffe.
“Hey get well soon Mark! Look forward to your full recovery,” wrote Simon Mayo, while Claudia Winkleman said: “Sending so much love.”
Ken Bruce added: “So sorry to hear this. Hope you’re back in action very soon.”
Radcliffe began his BBC career in 1983, and presented on both 5 Live and Radio 1 before he moved to Radio 2 in 2004.
While at Radio 1, he hosted the breakfast show as one half of duo Marc and Lard alongside Marc Riley.
Facts about tongue cancer
It can either affect the oral tongue, or the base of the tongue.
- A persistent red or white patch
- Pain when swallowing
- Unexplained bleeding
- A sore throat that does not go away
- A numbness in the mouth
These symptoms could also arise from a less serious condition.
According to Cancer Research UK, smoking, alcohol and infection with the HPV virus can increase a person’s risk of contracting tongue cancer.
Treatment in its early stages includes surgery to remove the cancer, and potentially a procedure to remove lymph nodes in a patient’s neck.
Radiotherapy and chemotherapy can also be used in both early and advanced stages of the disease.