Depp brought the legal action, claiming he had been unfairly slurred based on accusations of domestic violence by Heard during their five-year romance.
The movie star responded to last November’s ruling by seeking permission to appeal, suggesting the judge had failed to properly assess the evidence and calling for a full re-trial.
If successful, Depp’s case would move on to a full appeal hearing, with the possibility of new evidence being introduced that was absent from last summer’s libel trial.
But if his appeal bid is rejected Mr Justice Nicol’s ruling would continue to stand, with Depp facing a sizeable damages and costs bill as well as a tarnished reputation.
Depp denied being violent to his actress wife, insisting she had mounted a “hoax” against him throughout their volatile romance.
But Heard told the court she had been left bloodied and bruised repeatedly at the hands of Depp when he was intoxicated and consumed by a “monster” inside.
In his ruling, Mr Justice Nicol favoured the evidence of Heard and her friends, finding that she had on three occasions been left fearing for her life due to Depp’s abuse.
Seeking to appeal the decision, Depp’s legal team applied to introduce new evidence which they say undermines Heard’s credibility as a truthful witness.
Andrew Caldecott QC, for Depp, said they believe Heard told a “calculated and manipulative lie” in a pledge to donate her $7 million divorce settlement to charity, suggesting the money has not been paid.
He said there was a “striking lack of criticism” of Heard in the libel judgment, arguing: “Had the truth about the charity claim emerged at the trial, it would have materially affected the Judge’s consideration of Ms Heard’s evidence as a whole”.
Lawyers for News Group Newspapers, publisher of The Sun, argued the evidence should not be admitted as it could have been gathered before the original trial, countering that it is “peripheral” and would not have had an impact on the outcome of the case.
“Given the wealth of evidence”, they argued, “it is plainly wrong to suggest that the information that Ms Heard had not yet finished paying $7 million to charity would have made the slightest difference to the outcome of this case.”
They argue the pledge to donate the money is truthful, and a plan is in place to honour the promise over ten years.
Lord Justice Dingemans is due to hand down the ruling on permission to appeal at 10am.