The pair, dressed in camouflage fatigues, have their right arms raised and hands out straight in the image.
The Union Flag is inscribed with "Invicta Loyal", the name of a Kent-based Rangers supporters club, and below it hangs the Ulster Banner, a loyalist version of the Northern Ireland flag.
It has been suggested 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.
However, 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 was at least three-and-a-half years old and has been circulating online 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.
Drew Robertson, of the Rangers Supporters Association, said: "I am 100% certain that the salute performed by these soldiers is not in any way showing support for the Nazis - they are doing a Red Hand of Ulster salute.
"The Red Hand of Ulster salute is ill-advised. People have compared it to the Nazi salute and it could be misinterpreted as such - it looks similar - so I'd ask people to desist from doing it."
John McMillan, former general secretary of the Rangers Supporters Association, added: "It is something that fans should really avoid doing as it can be misinterpreted as a Nazi symbol and offend people. I do not think it is something we should be associated with and I do not think the club would welcome it."
An MoD spokesman said: "Individuals are free to join political parties but are expected to abide by our values and standards in all they do.
"Regular Service personnel are not to take any active part in the affairs of any political organisation, party or movement and they are not to participate in political marches or demonstrations.
"All those who are found to fall short of the Armed Forces' high standards or who are found to have committed an offence under the Armed Forces Act are dealt with administratively (up to and including dismissal) or through the disciplinary process."
The emergence of the photo in the run-up to Remembrance Sunday and the salute's association with Nazism has provoked disgust and outrage.
John Mann, chairman of the Commons' All Party Group Against Anti-Semitism, said: "Whether they call it a 'Red Hand salute' or whatever makes absolutely no difference - this is one of the most offensive gestures they could make in the modern world."
Last month investigations were launched into the conduct of military personnel and allegations of sectarian singing at a Rangers match at Ibrox.
Hundreds of members of the Royal Navy, Royal Marines, Army, and Royal Air Force took part in the Armed Forces Day event staged at half time on September 28, and footage later emerged of soldiers in uniform interacting with fans while singing about proscribed loyalist organisations.