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The soldiers, who have not yet been charged, remain on active duty while the Defence Serious Crimes Unit investigates the incident.
The allegations, first reported by The Daily Mail and The Guardian, concern claims the elite soldiers used excessive force when dealing with the suspected militant while deployed inside Syria and should have arrested the man instead.
The troops reportedly believed he had intended to carry out a suicide attack.
The Daily Mail, citing anonymous SAS sources, said a primed suicide vest was found nearby but the suspect was not wearing it when he was killed.
Following an investigation by the Defence Serious Crimes Unit, military chiefs reportedly sent files recommending murder charges against the five soldiers to the Service Prosecuting Authority.
Asked about the allegations, a spokesperson for the Ministry of Defence said: "We hold our personnel to the highest standards and any allegations of wrongdoing are taken seriously.
"Where appropriate, any criminal allegations are referred to the Service Police for investigation."
British special forces soldiers have been deployed in Syria over the last decade in the fight against Islamic State.
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Secret inquiry
It comes after the chair of a public inquiry into allegations of war crimes by British armed forces in Afghanistan decided the inquiry would be held partly in secret.
The allegations of extrajudicial killings are part of an investigation by the BBC and The Times which claimed rogue SAS units executed innocent civilians during a campaign of night raids set up to capture Taliban fighters.
Evidence submitted to the inquiry claimed as many as 80 people were killed in suspicious circumstances by three out of four SAS squadrons between 2010 and 2013.
The documents highlighted the high kill rate of the squadrons, with one soldier shooting 35 people dead in a single six-month tour.
Sky News is part of a number of media outlets that submitted challenges to the application for restrictive orders from the MoD and Royal Military Police (RMP), which is accused of failing to investigate the claims.