The anticipated showdown, which could take place as early as next week, comes shortly after Tesla unveiled its all-electric Cybertruck.
During the promotional event in Los Angeles last week, which turned into a somewhat embarrassing debut for the company's first electric pickup truck, Elon Musk showcased a promotional video of the vehicle engaging in a tug-of-war battle against a Ford F-150.
The 16-second clip, which has since been viewed more than 12 million times, shows Tesla's Cybertruck hardly breaking a sweat during the display, as it comfortably pulls a screeching Ford F-150.
However, critics of the demonstration have since argued the test was unfavorable to Ford.
Sundeep Madra, vice president at Ford X, sent a message via Twitter to Musk on Monday, apparently urging the Tesla CEO to send over a Cybertruck for an "apples to apples" tug-of-war test.
In a tweet directly replying to the Ford X vice president, Musk replied: "Bring it on."
A Ford spokesperson was not immediately available to comment when contacted by CNBC Tuesday morning.
Astrophysicist Neil deGrasse Tyson — who has previously suggested that he believes Musk is more important than Jeff Bezos, Steve Jobs and Mark Zuckerberg — was one of those skeptical about the legitimacy of the original tug-of-war test.
Tyson said the stunt may not have demonstrated the pickup truck's engine power, challenging Musk to "fully load" the Ford F-150, "giving highest traction to its rear wheels, then try to drag that up the hill."
Musk agreed to do that next week.
He has since suggested the next tug-of-war test will be available to watch via webcast.
On Sunday night, Musk said Tesla had received 200,000 orders for its Cybertruck, despite a botched demo that led to a broken window and sparked meme's making fun of the vehicle's steel trapezoid shape.
At the unveiling, Musk tried to showcase the Cybertruck's durability by having the company's design chief hurl a metal ball at one of the Cybertruck's armored glass windows.
The window ended up cracking. Then another ball was thrown at another window and it shattered, too.
— CNBC's Lora Kolodny contributed to this report.