The video then shows an image of Reynolds.
"Think before you vote. But definitely see The Front Runner. I'm not Ryan Reynolds and I approved this message."
In The Front Runner, which opens in theaters on November 21, Jackman portrays Colorado Senator Gary Hart who was considered the frontrunner for the 1988 Democratic presidential nomination.
However, an extramarital affair sidelined his bid in what was considered the first time tabloid journalism and political journalism merged.
Jackman responded with his own social media jab at Reynolds.
He posted a video on his Twitter account showing his dog defecating on a photo of Reynolds.
Last year, Jackman decided he would no longer portray Wolverine in the Marvel Comics series.
The 50-year-old Aussie revealed that Reynolds urged him to star in a crossover Deadpool-Wolverine film.
The Canadian has played Deadpool opposite Hugh before, having debuted a vastly different version of the character in the 2009 movie X-Men Origins: Wolverine.
While that film was a critical failure and was widely parodied by fans - and often by Ryan - it spawned a bromance between the two actors that has sparked regular comedic interplay between them online.
The 2016 Deadpool reboot was a box office smash however, with their online banter a significant component of the film's unusual publicity campaign.