The 60-year-old, who began his BBC career in 1983, revealed his diagnosis to listeners while presenting The Folk Show on Wednesday 3 October.
"I'm said to say that I've got some cancerous tongue and lymph node issues," he said, adding that 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," he reassured fans.
On Twitter he added: "All being well I'll be back in action in the New Year – or sooner if I feel well enough – but I will be back, you can depend on it, I just can't say exactly when that will be. So you won't hear from me for a while now and therefore... well... Merry Christmas I guess. Life eh?"
Radcliffe confirmed that he would broadcast his Thursday 6 Music show with Stuart Maconie on 4 October, as usual, before taking a break. Ricky Ross will present the Radio 2 Folk Show for the next few weeks.
A BBC spokesperson said: "Everyone at Radio 2 and 6 Music wishes Mark well and we look forward to welcoming him back when he's ready."
Fellow BBC presenters also sent Radcliffe their best wishes for his recovery.
"Hey, get well soon Mark! Look forward to your full recovery," Simon Mayo wrote, while Claudia Winkleman said: "Sending so much love."
"So sorry to hear this," Ken Bruce wrote. "Hope you're back in action very soon."
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