The pair, dressed in camouflage fatigues, have their right arms raised and hands out straight in the image, published in The Mail on Sunday (MoS).
However, their gesture could be an apparent show of solidarity with Northern Irish loyalists.
The Union Flag is inscribed with the words “Invicta Loyal”, the name of a Kent-based Glasgow Rangers supporters club, and below it hangs the Ulster Banner, a loyalist version of the Northern Ireland flag.
There is a suggestion the soldiers may be making a “Red Hand of Ulster” salute, once used by Rangers fans as a sign of allegiance with Northern Ireland Loyalists, the MoS said.
Democratic Unionist MP Sammy Wilson said: “This is not a salute made by marching bands (in Northern Ireland). It is not clear what the context is, but I do not think it can be explained away as some type of Red Hand of Ulster salute.”
Jeffrey Donaldson MP of the Democratic Unionist Party added: “Anyone who would imitate a Nazi salute in front of the Union Flag or wearing the uniform dishonours these flags and the British army.
“Any symbolism associated with fascism has no place in the unionist tradition or any regiment of the British army.
“We do not know to which regiment these soldiers belong but the Royal Irish Regiment, with which I am associated, has a proud record of fighting fascism.”
The Ministry of Defence (MoD) confirmed the men were serving soldiers when the photo was taken, reportedly at an army base in Afghanistan. The MoD said the image is at least three and a half years old and has been circulating on online forums for some time.
The MoD said the incident has been investigated and “administrative action” was taken at the time, in which the soldiers were dealt with by their chain of command.
It is not known if they were dismissed.