At the unveiling of the futuristic electric truck last week, the Tesla boss showed a video of the Cybertruck in a tug-of-war contest with Ford's best-selling pickup.
The Ford appeared to be easily dragged by the Tesla, which Mr Musk claimed was all the more impressive because "it was uphill".
Analysis of the video by some motor journalists suggested the test was not entirely fair, as it appeared Tesla had pitted a rear-wheel drive model of the Ford F-150 against an all-wheel drive version of the Cybertruck.
Sundeep Madra, a vice president at Ford, tweeted an article questioning the fairness of the demonstration, together with the challenge: "Hey @elonmusk send us a Cybertruck and we will do the apples to apples test for you."
The Tesla CEO was quick to respond: "Bring it on."
In a separate conversation on Twitter, the billionaire also got involved in a discussion with astrophysicist Neil deGrasse Tyson about the physics involved in the tug-of-war contest between the Cybertruck and F-150.
"Electric vehicles are famously heavy - over both axles. It's all about the weight borne by spinning tyres," Mr DeGrasse Tyson wrote. "That's the source of the traction, not the engine power."
Mr Musk replied that electric motors have "insane torque" and that even if both trucks were fully loaded, the Cybertruck would still win. "Physics is the law, everything else is a recommendation," he concluded.
The physicist suggested testing this theory, to which Mr Musk agreed, together with a promise to live stream the event on the internet.
Another claim made by Tesla during the Cybertruck unveiling event was that it is faster than a Porsche 911, capable of accelerating from 0-60mph in just 2.9 seconds.
It comes with the official tagline: "Better utility than a truck with more performance than a sports car."