Lineker's son, who is now aged 20, successfully beat a rare form of the illness when a baby.
Writing on Twitter yesterday (August 1), the BBC Olympics presenter and ex-England footballer said: "I'm about to [re-tweet] three tweets that were sent to my son that make me feel physically sick. Culprit has deleted his account."
The messages included the comments "heard ya Leukaemia , pity ya didn't die" and "cheers for the retweet cancer boy ! You f*****g gent".
Another tweet read: "Haha your dad's a washed up piece of s**t Leukaemia boy."
Lineker claimed that the offending messages came from a "tiny mindless minority" and told his followers that his son has not let it bother him.
George wrote on Twitter: "Don't waste your time having a dig at someone who it doesn't bother in the slightest, if it bothered me I'd have come off Twitter a year ago."
This is the latest case of a celebrity highlighting Twitter abuse, following cases involving Olympic diver Tom Daley and BBC presenter Helen Skelton.