The pair are dressed in camouflage fatigues in the image, published yesterday in a Sunday paper.
Their gesture could also be an apparent show of solidarity with Northern Irish Loyalists, although Unionist MPs dismissed this suggestion.
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 flag of Northern Ireland.
There is a suggestion the soldiers may be making a “Red Hand of Ulster” salute.
But 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 when 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.”
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, 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.