Jones, from Chatham in Kent, joined so-called Islamic State after converting to Islam and traveling to Syria in 2013.
Her death was first reported by The Sun.
The BBC’s security correspondent Frank Gardner said, Jones had been a useful propaganda agent for IS on social media and her death would be “significant”.
Whitehall officials have declined to comment publicly. However, they have not denied the story, and US sources are confident she was killed in an unmanned drone strike in June, our correspondent added.
Posing with weapons
Previously a punk musician, she had been used to recruit western girls to the group and posted threatening messages to Christians in the UK.
Born in Greenwich, London, Jones also encouraged individuals to carry out attacks in Britain, offering guidance on how to construct home-made bombs.
She used her Twitter account to provide practical advice on how to travel to Syria and shared pictures of herself posing with weapons.
By BBC security correspondent Frank Gardner
The death of Sally-Anne Jones, if confirmed, will make little or no tactical difference to the military fortunes of so-called Islamic State on the battlefield.
Despite posing online variously with a Kalashnikov and a pistol, and reportedly “leading a battalion of jihadist women”, her value to the group was iconic, rather than military.
But in this role she was definitely judged to be dangerous.
Along with her late husband, Junaid Hussain, killed in a drone strike in 2015, she maintained a stream of hostile online propaganda aimed at the West.
This included luring western female recruits to the self-declared IS caliphate, encouraging attacks in the West and threatening to kill non-Muslims.
She is believed to have been involved in planning previous attacks in the West, including a plot to assassinate the Queen and Prince Philip in 2015.
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Fears for young son
Jones’s husband, Hussain, was a computer hacker for IS and was regarded as a “high value target” before his death.
In 2015 the then Prime Minister David Cameron said Hussain had been planning “barbaric attacks against the West”, including terror plots targeting “high profile public commemorations”.
News of Jones’s death had not been made public amid fears that her 12-year-old son, Jojo, may also have been killed, according to The Sun.
Major General Chip Chapman, the former Ministry of Defence head of counter terror, said Jones would have been a “significant” target as a result of her alliance with Hussain and her role in recruiting IS fighters.
Responding to reports her son was killed in the strike, he added: “It is a difficult one because under the UN Charters he is under the age of what we would classify as a soldier.”
Azadeh Moaveni, a journalist and author of the book Lipstick Jihad, told the BBC Jones had been one of the most “iconic” recruiters for IS because she helped the group to project the idea it could “get into the very the reaches of British society”.
A Foreign Office spokeswoman said: “We do not comment on matters of national security.”