Some Liverpool supporters were heard using anti-gay language against the 20-year-old on-loan Chelsea midfielder Gilmour at Carrow Road last Saturday.
The German said such songs were a "complete waste of time" and that he did not want to listen to them again.
After Saturday's match, the Merseyside club issued a statement saying: "Liverpool FC is disappointed by reports that a small section of our supporters were heard singing offensive songs during Saturday's fixture with Chelsea at Anfield.
"The club strongly condemns these actions and will be investigating all reported allegations of abusive and/or discriminatory behaviour with the relevant authorities to identify perpetrators.
"LFC is committed to tackling abusive and discriminatory behaviour, which has no place in football or society."