Trump on Saturday said the outcome might have been different if the synagogue "had some kind of protection" from an armed guard, while Pennsylvania's Democratic Gov. Tom Wolf, up for re-election, noted that once again "dangerous weapons are putting our citizens in harm's way".
"This falls under hate crime", he said.
"This was an antisemitic act". "I don't know what to tell you". "Our minds cannot comprehend the cruel hate and the twisted malice that could cause a person to unleash such awful violence during a baby naming ceremony".
"We are praying for the families of the victims".
"These senseless acts of violence are not who we are as Americans".
Authorities say no children are among the 11 dead.
The gunman, who has been described as a white man with a beard, is still in the building but has surrendered to police and is in custody.
Bowers made threats on social media to the Hebrew Immigrant Aid Society, an organisation that supports Jewish refugees, before his killing spree.
The Federal Bureau of Investigation took the lead in the investigation, "as this falls under a hate crime", Pittsburgh Public Safety Director Wendell Hissrich told reporters near the synagogue.
Finkelstein said local synagogues have done "lots of training on things like active shooters, and we've looked at hardening facilities as much as possible".
Pittsburgh police Commander Jason Lando said: "Do not come out of your home right now, it is not safe".
President Donald Trump told reporters at Joint Base Andrews in Maryland that the shooting was a "terrible, awful thing".
In 2010, Tree of Life Congregation - founded more than 150 years ago - merged with Or L'Simcha to form Tree of Life (asterisk) Or L'Simcha.
"When people do this they should get the death penalty", he said. "They should pay the ultimate price, I've felt that way for a long time". "This has to stop".
According to photos and videos from the scene, police and emergency services, as well as the SWAT team have been deployed to the area around the attack.
"Everyone is just shaken up and upset", Rabner said. "He didn't see us, thank God". "Avoid the Squirrel Hill/Shadyside area", the university police tweeted.
Michael Eisenberg, past president of the synagogue, told local television the door would typically have been open on Saturdays with religious services going on.
"The world is a violent world", he said, calling the suspect "a madman, a whacko".
According to initial reports, multiple people have been shot, but the exact number was not immediately known.
A gunman opened fire on Jewish worshippers at a synagogue in the United States city of Pittsburgh on Saturday, reportedly killing at least four people and injuring half a dozen others before being taken into custody.