News magazine Der Spiegel, which didn't cite its sources, said the suspect is a 27-year-old man from Saxony-Anhalt state, where Halle is located.
Media quoted investigators as revealing they have a video that the assailant apparently filmed with a helmet camera and posted on an internet site.
A shooting in the eastern German city of Halle has left at least two people dead.
Police had earlier said the "perpetrators fled in a car" before saying later that one suspect had been caught.
Officers were out in force across Halle, a city of 240,000.
The suspect is now on the run, with authorities advising residents to "remain in their homes or to find a safe place", police said.
They did not want to be named due to security concerns.
The footage shows the man, wearing military gear and a helmet, getting out of a auto and firing several times, stopping to reload each time. It wasn't clear whether that incident was related to the shooting in the city.
Images aired by German news outlets showed a body lying about 30 yards away from the synagogue, according to the Associated Press. He then flung an explosive at the gate of the Jewish cemetery but failed to break it open.
Police added several shots were fired and the alleged shooter fled in a vehicle.
"God has protected us, the whole maybe took five to ten minutes", he said. Yom Kippur, also known as the Day of Atonement, is a period of time for people to ask for forgiveness for their sins and transgressions.
"(The attacker) shot at least once in the shop, the man behind me must be dead.
"The others looked for the back entrance".
"I am shocked and don't know how I will feel in this city in the coming days and weeks".
"Together with you, I would like to express my sympathies to the German police forces, the German people and the German Jewish community", European Parliament President David Sassoli said, leading the tribute.
Federal police also reinforced security at train stations and airports in central Germany and on routes to Poland and the Czech Republic.
Police increased their presence outside synagogues in Halle and nearby areas after the attack, according to the report.
"We have a deployment in Halle", the native police stated on Twitter.
"Several shots were fired", Halle police tweeted. Scores of police vans and a helicopter could be seen on the edge of the small town a few hours later.
German chancellor Angela Merkel joined a vigil held at Berlin's New Synagogue after the attack.