Reynolds is known for jokingly trolling The Greatest Showman on social media.
The actor recently tweeted a mock political ad about Jackman, 50, and his latest movie, The Front Runner, in which he portrays one-time presidential candidate Gary Hart.
By Reynolds' reckoning, Jackman "has fooled some people into thinking he deserves an award".
"But before voting begins, some people should consider these facts," the video says, before going on to note that "Hugh Jackman isn't his real name."
It's actually "Hugh Michael Jackman," says an in-character Reynolds in the video, as a birth document highlighting the actor's middle name appears on the screen.
"Hugh Michael speaks with a charming accent but he's actually from Milwaukee," he adds, suggesting that Jackman contributed to the nation's unemployment rate by walking "off the job on Wolverine".
"Is this the type of amazing actor and nice person we want delivering a preachy liberal speech this award season?" Reynolds says.
"Or maybe there's a real foreigner from a bordering country who would really love it. A lot."
At that moment, an image of Reynolds, who was born in Canada, is seen.
"I'm not Ryan Reynolds and I approve this message," the video continues.
Jackman then shot back on Twitter with a video of his own.
In the clip, captioned "HIGH ROAD", the Wolverine actor is seen cleaning up dog droppings from atop a photo of Reynolds.
