The pair have traded places as the world’s richest person in recent years, using their vast wealth to fund their private space companies.
Mr Musk’s SpaceX won a lucrative contract from Nasa earlier this year to develop its Starship craft for the 2024 Artemis mission, which aims to return humans to the Moon.
Mr Bezos’s Blue Origin firm missed out on the funding, prompting them to file a 50-page protest to the Government Accountability Office that accused Nasa of “moving the goalposts at the last minute”.
Blue Origin described the bidding process as “flawed”, to which Mr Musk replied: “Can’t get it up (to orbit) lol.”
On Wednesday evening, the SpaceX boss built on this jab by tweeting: “If lobbying and lawyers could get u to orbit, Bezos would be on Pluto rn.”
It is the latest in a series of barbs traded by the billionaires in recent years, as each firm battled to beat the other to major space flight milestones.
In 2015, Mr Bezos described the first successful launch and landing of Blue Origin’s New Shepard space vehicle as “the rarest of beasts”, to which Mr Musk responded: “Not quite ‘rarest’. SpaceX Grasshopped rocket did six suborbital flights three years ago and is still around.”
In 2019, Mr Musk shared a mocked up image of Blue Origin’s lunar lander, changing the name from Blue Moon to Blue Balls.
Last month, just days after a 10-minute flight to the edge of space aboard Blue Origin’s New Shepard rocket, Mr Bezos published an open letter to Nasa calling for competition to be restored for the Artemis mission.
He also offered to cover large parts of the costs in the development of a lunar lander.
“I am honoured to offer these contributions and am grateful to be in a financial position to be able to do so,” he wrote.
Mr Musk later replied indirectly, tweeting: “Just want to say thanks to those in government who fight hard for the right thing to happen, despite extreme pressure to do otherwise. Therein lies the core goodness of the American state.”