The man has been identified by the FBI as 44-year-old Malik Faisal Akram, originally from Blackburn, Lancashire.
He was killed in a “shooting incident” after the FBI entered the building at Congregation Beth Israel in Colleyville, Texas, on Saturday.
All four hostages were unharmed.
During the incident – which saw police first called at 11am and ended with the release of the final hostages at around 9pm – the man could be heard ranting on a livestream in what appeared to be a British accent.
On Sunday, the Foreign Office confirmed he was British and the Metropolitan Police said counter-terror officers were in contact with US authorities and colleagues from the FBI.
Foreign Secretary Liz Truss condemned the “act of terrorism and anti-semitism”, while the British Ambassador to the United States Karen Pierce said British authorities are providing “full support to Texas and US law enforcement agencies”.
The man is said to have demanded the release of a Pakistani neuroscientist who was convicted of trying to kill US army officers in Afghanistan.
Speaking to reporters after the incident, FBI special agent in charge Matt DeSarno said they believed the man was “singularly focused on one issue and it was not specifically related to the Jewish community”, and added they will continue to “work to find motive”.
Confirming that the hostage-taker had died, he said there would be “an independent investigation of the shooting incident”.
A spokesman for the Foreign, Commonwealth and Development Office said: “We are aware of the death of a British man in Texas and are in contact with the local authorities.”
One hostage who had been held was released during the stand-off and the three others got out when an FBI Swat team entered the building, authorities in the US said.
The Metropolitan Police confirmed officers from counter-terrorism policing are “liaising with US authorities and colleagues from the FBI” regarding the incident.
Ms Truss tweeted: “My thoughts are with the Jewish community and all those affected by the appalling act in Texas. We condemn this act of terrorism and anti-semitism.
“We stand with US in defending the rights and freedoms of our citizens against those who spread hate.”
Ms Pierce said: “The UK & US stand shoulder to shoulder in defiance of terrorism and in defence of the fundamental rights and freedoms of our citizens.”
Earlier, US law enforcement officials said the man had demanded the release of Aafia Siddiqui, a Pakistani neuroscientist suspected of having ties to al Qaida, who is in prison in Texas.
The officials revealed that he said he wanted to be able to speak to her.
While a number of people were reported to have heard the hostage-taker refer to Siddiqui as his “sister” on the livestream, her brother is said not to have been involved.
John Floyd, board chairman for the Houston chapter of the Council on American-Islamic Relations and long-time legal counsel for Siddiqui’s brother, said his client was not the person responsible for the “heinous” incident.
Mr Floyd condemned what had happened as “wicked”, and said the person involved “has nothing to do with Dr Aafia, her family, or the global campaign to get justice for Dr Aafia”.