Malik Faisal Akram, who took four people hostage in the US, was investigated in 2020 by UK services.
A British man who took four people hostage at a synagogue in Texas had been investigated by MI5.
Malik Faisal Akram, from Blackburn in Lancashire, was the subject of an investigation in late 2020 but by the time he flew to the US he was assessed to be no longer a risk.
The four people held hostage at the synagogue in Colleyville near Dallas were eventually freed unharmed, after a 10-hour siege.
Akram, 44, was shot dead by police.
Two teenagers were arrested in England as part of the investigation but their ages and genders have not released.
US President Joe Biden has described the hostage-taking as an “act of terror”.
BBC security correspondent Frank Gardner said Akram had been on the British security service’s watchlist as a “subject of interest” in 2020.
The siege began at around 11:00 local time (16:00 GMT) on Saturday, when police were called to the Congregation Beth Israel synagogue.
Akram gained initial access to the synagogue during the service by claiming to be a homeless man, according to a police source quoted by CBS.
One hostage was released after six hours, while the other three – including the synagogue’s rabbi – escaped several hours later.
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Both Greater Manchester Police and the Metropolitan Police have said they are in contact with the US authorities over the investigation.
Akram is thought to have arrived in the US via New York’s JFK International Airport two weeks ago, according to police sources, and he is believed to have bought weapons used in the incident “on the street” after his arrival.
During the standoff Akram was heard demanding the release of Aafia Siddiqui, a Pakistani neuroscientist who is serving an 86-year prison sentence in nearby Fort Worth, Texas, over attempts to kill US soldiers in Afghanistan.
Siddiqui has distanced herself from his actions, issuing a statement through a lawyer.