The Colleyville Police Department said it had removed residents from the area around the Congregation Beth Israel as it deployed SWAT teams.
US President Joe Biden has been briefed on the "developing hostage situation", his press secretary said, and FBI agents were negotiating with the man, CNN reported.
Israeli Prime Minister Naftali Bennett said on Twitter that he was monitoring the situation and was praying for the safety of the hostages.
No injuries have been reported inside the building in the Dallas-Fort Worth suburb, according to the Dallas Morning News.
The man on Saturday could be heard having a one-sided conversation in what appeared to be a phone call during a Facebook live stream of the Reform Jewish synagogue's Shabbat service.
The live stream cut off about 3pm.
Before the live stream was ended, the man could be heard ranting and talking about religion and his sister, the Fort Worth Star-Telegram reported.
The man could be heard repeatedly saying he did not want to see anyone hurt and he believed he was going to die, the newspaper said.
Barry Klompus, a member of the congregation since it opened in 1999, said he tuned into the live stream.
"It was horrible listening and watching, and it's that much more horrible not knowing," Klompus said in a telephone interview.
Though he was not able to clearly understand what the man wanted, Klompus believes the man wanted to talk to his sister.
A US official briefed on the matter told ABC News the hostage-taker was claiming to be the brother of Pakistani neuroscientist Aafia Siddiqui, who is serving an 86-year US prison sentence after her 2010 conviction for shooting at soldiers and FBI agents, and demanding she be freed.
But authorities had not yet confirmed his identity, the official told ABC News.
Siddiqui is being held at a federal prison in the Fort Worth area.
The president of the Union for Reform Judaism, Rabbi Rick Jacobs, said on Twitter the union was "very grateful to law enforcement who are working to free the hostages".
The Anti-Defamation League, a Jewish advocacy group, said it was aware of the standoff, and CAIR, a US Muslim advocacy group, condemned the man's actions.
"This latest anti-Semitic attack on Jewish Americans worshipping at a synagogue is an act of pure evil," CAIR said in a statement.
Klompus said he did not know of any significant previous threats to the congregation.
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