Köhler has asked for “supplementary information,” the Spiegel report said.
The law, which critics argue would block access to other, innocent sites and therefore amounted to censorship, could breach Germany's constitution, experts believe.
The law was written under the previous “grand coalition” government between Angela Merkel's Christian Democrats (CDU) and the centre-left Social Democrats and was pushed by then CDU Family Minister Ursula von der Leyen.
Merkel's party and their new partners in government, the pro-business Free Democrats – who opposed the measure – agreed during coalition negotiations last month not to put the law into practice.
But because it had already been passed by both houses of the German parliament, it could not simply be dropped. Köhler's refusal to sign it means it is now effectively stalled until the new government finds a constitutional way to kill it.
According to a Saturday report in business magazine Wirtschaftswoche, Interior Minister Thomas de Maizière and Justice Minister Sabine Leutheusser-Schnarrenberger had agreed to kibosh the law by ordering the federal police not to act upon it. However, that would leave the law hanging in place.
One possibility, the Spiegel report said, is that the parliament issue a regulation repealing the law, which Köhler can wait to sign when it crosses his desk.