Malik Faisal Akram, originally from Blackburn in Lancashire, was shot dead when the FBI entered the building in Texas on Saturday night.
US President Joe Biden branded the incident “an act of terror” and UK police are working with authorities in America on the investigation.
Greater Manchester Police (GMP) announced that officers from Counter Terror Policing North West had made two arrests in south Manchester on Sunday evening.
They said the teenagers, whose ages and genders they did not immediately confirm, remain in custody for questioning.
GMP said police forces in the region are liaising with local communities to put in place any measures to provide further reassurance.
The four hostages held at Congregation Beth Israel in Colleyville, Texas were unharmed.
Akram’s family said they were “absolutely devastated” by what had happened and “do not condone any of his actions”, according to a statement which had been shared on the Blackburn Muslim Community Facebook page.
The statement, attributed to Akram’s brother Gulbar who said he had been involved in negotiating from the UK with his sibling during the ordeal, added that the hostage-taker “was suffering from mental health issues”.
US officials believe Akram had a visa, arrived at John F Kennedy International Airport in New York around two weeks ago and bought a handgun used in the incident.
In an update to reporters on Sunday, Mr Biden said while he did not have all the details it was believed Akram had “got the weapons on the street”, adding: “He purchased them when he landed.”
He said there were “no bombs that we know of”, and that Akram is thought to have “spent the first night in a homeless shelter”.
Condemning what had happened, the statement from Akram’s family said: “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.”
Akram is said to have demanded the release of Pakistani neuroscientist Aafia Siddiqui, who was convicted of trying to kill US army officers in Afghanistan, and is in prison in Texas.
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”.
He said the FBI had been in contact with their legal attache offices in London and Israel for an investigation with “global reach”.
Foreign Secretary Liz Truss condemned the “act of terrorism and anti-semitism”, while the British Ambassador to the United States Karen Pierce said UK authorities are providing “full support to Texas and US law enforcement agencies”.