According to German newspaper Bild, Klopp is prepared to battle with Man City boss Pep Guardiola for the 23-year-old Bosnian international.
His Schalke contract expires in the summer, so the German club will be keen to cash in when the transfer window opens in January rather than let him leave on a free at the end of the season.
Having already signed Kolasinac's former teammate Joel Matip last summer, Klopp is sure to be favourite to do the deal
Source: DSG