"Please ignore the stupidity. Alive and well and happy and healthy...Still punching!" the 71-year-old Stallone, the hero of the popular "Rambo" and "Rocky" series, posted on his Twitter and Instagram handle.
Terming the social media reports of his brother's death as rumours, Frank, 67, said in a video he posted on Instagram: "What kind of sick demented cruel mind things of things like this to post? People like this are mentally deranged and don't deserve a place in society."
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Sylvester pointed out that the images purporting to show him suffering from an illness that made him lose his hair were likely from a scene he has filmed for one of films.
There was a spate of social media posts on Monday claiming that the actor had died of cancer.
This is not the first time that erronious messages of Sylvester's demise had flooded social media. In 2016, a British media news site had claimed he was dead, prompting him to upload a photo of himself with Russian boxer Sergey Kovalev on Instagram.
Sylvester's last post on social media was made on Sunday to promote his new film "Creed II".