Karius is out of favour at Anfield and is currently behind Alisson, Caoimhin Kelleher and Adrian in Jurgen Klopp’s pecking order.
Liverpool were actively looking to sell the 28-year-old in the recently concluded transfer window, but could not move him on.
Swiss top flight club Basel were interested in acquiring Karius’ services and approached the Merseyside-based club with an offer.
However, the transfer did not materialise as the German goalkeeper failed to agree personal terms with his suitors.
While Basel failed in their attempts to sign Karius in the summer, they remain interested in signing him, according to German daily Bild.
The Swiss giants will make another attempt to sign the former Mainz goalkeeper from Klopp’s side when the winter transfer window opens.
It is suggested that the biggest factor in the transfer happening is likely to be whether Karius wants to make the move to Basel.