British security services had assessed the gunman in a prior investigation and found he was no longer a threat at the time he travelled to the US.
British security services had recently investigated the man behind a hostage taking at a synagogue in Texas but deemed he posed no threat, media reports said.
US authorities identified the captor as British citizen Malik Faisal Akram, 44, who was shot dead in a 10-hour siege in the small town of Colleyville on Saturday.
Britain's MI5 domestic intelligence agency received information about Akram, who came from Blackburn in northwest England, in 2020, prompting an investigation, several news outlets said, citing unnamed government sources.
But it was shut down again after a little over a month due to lack of evidence that he was a threat, the report added.
The agency is now expected to review the investigation.
MI5 keeps tabs on around 3,000 suspects at any one time, and has investigated up to 40,000 individuals in total.
Known criminal record
The four hostages — including a respected local rabbi, Charlie Cytron-Walker — were all freed unharmed on Saturday night.
The release of hostages prompted relief in the United States, where the Jewish community and US President Joe Biden renewed calls to fight anti-Semitism.
However, the incident raised questions about why Akram, whose family said he had mental health problems and was known to have a criminal record, was allowed out of the country at the end of last year.
Several British media outlets also said that Akram was banned from a local court in Blackburn for remarks he made to staff about the September 11, 2001 attacks in the US on the day of the attack.
Meanwhile, Biden declined to speculate on Akram's motive but appeared to confirm reports that the hostage-taker was seeking the release of Aafia Siddiqui, a Pakistani neuroscientist known as "Lady Al Qaeda."
US Muslim groups have condemned the hostage-taking incident at a synagogue in the US state of Texas.