Malik Faisal Akram, a 44-year-old British citizen, arrived in the US at Kennedy Airport in New York on a tourist visa about two weeks ago, officials said.
He spent time in Dallas-area homeless shelters before the attack on Saturday in the suburb of Colleyville.
"Our understanding, and obviously we're still looking into this, is that he was checked against US government databases multiple times prior to entering the country, and the US government did not have any derogatory information about the individual in our systems at the time of entry," White House press secretary Jen Psaki said.
"We're certainly looking back ... what occurred to learn every possible lesson we can to prevent attacks like this in the future."
British media reported that Akram was investigated by the domestic intelligence service MI5 as a possible terrorist threat in 2020, but the investigation was closed after authorities concluded he posed no threat.
The stand-off in Colleyville, 50 kilometres northwest of Dallas, ended after the last of the hostages ran out of the synagogue and an FBI SWAT team rushed in.
Akram was killed, but authorities have declined to say who shot him, saying it was still under investigation.
President Joe Biden called the episode an act of terrorism.
Rabbi Charlie Cytron-Walker, who was among the hostages, told CBS he had let Akram into Congregation Beth Israel on Saturday morning because he appeared to need shelter.
Cytron-Walker said the man was not threatening or suspicious at first, but later he heard a gun click as he was praying.
One hostage was released hours later, and the rabbi and two others later escaped after Cytron-Walker threw a chair at the gunman.
During the stand-off, Akram could be heard on a Facebook live stream demanding the release of Aafia Siddiqui, a Pakistani neuroscientist who is suspected of having ties to al-Qaeda and was convicted of trying to kill US troops in Afghanistan.
The prison where Siddiqui is serving her sentence is in nearby Fort Worth.
A lawyer in Texas who represents Siddiqui said the prisoner had no connection to Akram.
