The day-long siege at the Congregation Beth Israel synagogue in Colleyville, about 25 kilometres northeast of Fort Worth, Texas, ended on Saturday night with all four hostages released unharmed and with the suspect dead.
It was not clear whether the gunman took his own life or was slain by members of the FBI hostage rescue team.
The suspect was reported to have demanded the release of a Pakistani-born scientist serving a federal prison sentence in the Dallas-Forth Worth area during 10 hours of tense hostage negotiations with authorities.
The FBI on Sunday identified the suspect as Malik Faisal Akram, 44, a UK citizen, but did not say how or when he entered the United States.
Akram's brother Gulbar posted on Facebook that the suspect, from Blackburn in northern England, suffered from mental illness and said the family had spent all night at the Blackburn police station "liaising with Faisal, the negotiators, FBI etc".
"There was nothing we could have said to him or done that would have convinced him to surrender," Gulbar wrote on the Blackburn Muslim Community's Facebook page.
He said the FBI was due to fly into the UK "later today".
"We would like to say that we as a family do not condone any of his actions and would like to sincerely apologise wholeheartedly to all the victims involved in the unfortunate incident," the brother wrote.
Later on Sunday in England, the Greater Manchester Police said officers from Counter Terror Policing North West "have made two arrests in relation to the incident" in Texas, adding the two teenagers detained "remain in custody for questioning".
US President Joe Biden, who was in Philadelphia with first lady Jill Biden for a visit commemorating the birthday of slain civil rights leader Martin Luther King Jr, called the hostage-taking "an act of terror".
Saturday's siege began in late morning, disrupting Sabbath services as the suspect took the rabbi and three other people hostage.
One hostage was released unharmed after six hours and the remaining three were freed later either shortly before or during the raid by the FBI's hostage rescue team.
SWAT teams from the Colleyville Police Department initially responded to the scene, and FBI agents arriving afterward opened contact with the suspect, who said he wanted to speak to a woman held in a federal prison.
The man, who had seemed to be carrying on a one-sided phone conversation during a Facebook live stream of the service, could be heard ranting and talking about religion and his sister, repeatedly saying he did not want to see anyone hurt, the Fort Worth Star-Telegram reported.
The hostage-taker claimed to be the brother of Pakistani neuroscientist Aafia Siddiqui, who is serving an 86-year federal prison sentence on a 2010 conviction for shooting at soldiers and FBI agents, and demanded that she be freed, a US official told ABC News.
Siddiqui is being held at a federal medical lock-up in the Fort Worth area. A lawyer representing Siddiqui, Marwa Elbially, told CNN the man was not Siddiqui's brother and Siddiqui's family condemned his "heinous" actions.
Although the Texas hostage situation appeared to be an isolated incident, synagogues in New York and elsewhere around the country ramped up security in response and officials condemned anti-Semitic acts.
"What happened yesterday at Congregation Beth Israel is a reminder that we must speak up and combat anti-Semitism and hate wherever it exists," US Vice-President Kamala Harris said in a statement.Â
