The newspaper, along with Thomas, tried to have the injunction overturned last week in the court but Mr Justice Eady ruled against the newspaper, arguing that there was "no suggestion of any legitimate public interest in publishing such material".
Today's challenge from the Sun follows an image of the footballer – known in court only as "CTB" – being printed on the front page of Scottish national the Sunday Herald yesterday.
Responding to the Herald's naming of the player, David Cameron said this morning that the UK's current privacy law is "unsustainable" and "not fair on the newspapers if all the social media can report this and the newspapers can't".
Speaking on ITV1's Daybreak, Cameron said: "It is rather unsustainable, this situation, where newspapers can't print something that everyone else is clearly talking about.
"But there's a difficulty here because the law is the law and the judges just interpret what the law is."
He added that he knew the identity of the footballer, "like everyone else".
A spokesman for the Sun said today: "Following publication on the front page of a Scottish newspaper it is clear this injunction is unworkable.
"When the prime minister says on breakfast television that he knows the identity of the footballer, it is time for the courts to do the right thing and end a situation where readers of some newspapers but not others are allowed to know the worst kept secret in the country.
"We have asked our lawyers to make an application at the high court and await the outcome with great interest."
The newspaper argues that the Herald's identification of the player will make it "virtually impossible for the footballer's highly-paid lawyers to argue his name is not already in the public domain".
An urgent parliamentary question on the issue of injunctions has been accepted and will be put to the culture secretary, Jeremy Hunt this afternoon.
At 3.30pm culture, media and sport select committee chairman John Whittingdale will ask Hunt to set out the Government's position on "granting and enforcement" of privacy orders.
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