The Süddeutsche Zeitung said the bomber had fled to Germany after he was released from prison in Turkey in 2001.
Citing "security experts," the newspaper said he had appealed for political asylum. German authorities rejected the request because they determined he had not been politically persecuted and because he had been actively involved with a radical Turkish Marxist group.
He was instructed to leave Germany, but went underground and disappeared.
It was this group, the Revolutionary People's Liberation Front (DHKP-C), claimed responsibility for Friday's bomb attack at the embassy, where a Turkish security guard was killed.
The group, which is fiercely anti-US and is blacklisted by Washington and the European Union as a terrorist organisation, identified the bomber as Alisan Sanli.
Turkish authorities have said he was previously jailed for his involvement in an attack on an Istanbul military compound in 1997.
German authorities had not put him under surveillance because he was not thought to pose a threat, the newspaper reported.