Musk even raised the notion of selling more of his Tesla Inc. shares, which would require him to pay capital gains taxes.
Sanders, 80, who unsuccessfully sought the Democratic Party presidential nomination in 2016 and 2020, on Saturday tweeted that "we must demand that the extremely wealthy pay their fair share." He didn't mention Musk by name.
Musk, whose $286 billion fortune has made him a target for supporters of a wealth tax, tweeted: "I keep forgetting that you're still alive."
In a subsequent tweet early Sunday, Musk asked: "Want me to sell more stock, Bernie? Just say the word…"
Musk was still firing off tweets 11 hours later, saying Sanders "is a taker, not a maker."
Musk offloaded almost $7 billion worth of Tesla stock last week, helping push the automaker's shares down 15 percent. The stock fell as much as 2.9 percent before the start of regular trading on Monday.