The incident happened at Congregation Beth Israel in Colleyville on Saturday.
The man could be heard ranting, in what appeared to be a British accent, on a livestream and demanding the release of a Pakistani neuroscientist who was convicted of trying to kill US army officers in Afghanistan.
Police were first called to the synagogue at about 11am.
One hostage who had been held was released during the hours-long stand-off and three others got out at about 9pm local time when an FBI Swat team entered the building, authorities said.
The hostage-taker was killed and the FBI said a team would investigate “the shooting incident”.
A spokesman for the Foreign, Commonwealth and Development Office said on Sunday: “We are aware of the death of a British man in Texas and are in contact with the local authorities.”
FBI special agent in charge Matt DeSarno said they have been in contact with their legal attache offices in London and Israel.
He told reporters: “Our investigation will have global reach. We have been in contact already with multiple FBI legats to include Tel Aviv and London.”
Mr DeSarno said they believe 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”.
All four hostages were unharmed and were due to be reunited with their family after the incident, he added.
Earlier, 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 currently in prison in Texas.
The officials said he said he wanted to be able to speak to her.