"For us, he was a blank slate," said a spokeswoman of the Federal Criminal Police Office, which handles cases related to terrorism.
An interior ministry spokeswoman also said that there has been no indications suggesting an Islamist link to the suspect.
The 29-year-old suspect was sentenced to two years and three months for burglaries in the southwestern city of Mainz and in Baden-Wuerttemberg state farther south, and jailed in 2016.
"He served a year in Germany before being expelled to France," a spokesman from Baden-Wuerttemberg's interior ministry told AFP.
According to the Tagesspiegel newspaper, the man broke into a dentist practice in Mainz, Rhineland Palatinate state, in 2012, making away with cash, stamps and gold used for teeth fillings.
Four years later, he hit a pharmacy in the Lake Constance town of Engen, Baden-Wuerttemberg, pocketing cash.
German authorities were on the lookout for the fugitive "along the Rhine" river region, the ministry spokesman said.
"But at the moment we do not believe that he has crossed into the country," he added.
The gunman opened fire Tuesday evening at the famed Strasbourg Christmas market, which draws thousands of visitors every year.
The shooting left two people dead and 13 wounded.
French authorities said the attacker had been on their list of extremists and "is actively being hunted by security forces".
Meanwhile, German police said they have released three people earlier detained following tipoffs from the public following the Strasbourg attack.
The three were in a taxi with French licence plates which was halted on the A1 motorway close to the city Bremen, a police spokesman in Delmenhorst told national news agency DPA.
Police said they have been freed after investigations ruled out any connection to the attack in Strasbourg.
Police also denied an earlier report that one of the three was masked.