The German footballer, who signed for the Reds from Mainz 05 in the summer transfer window, is yet to make his competitive debut for the club due to a broken hand.
According to the Daily Star, during his time away from the pitch, he got to know former Hollyoaks actress Davis, who is expecting a baby boy with ex-boyfriend Jeremy McConnell.
However, Karius has rubbished the rumours by tweeting: "It's interesting being linked with a person, you've never heard of before. Please stop inventing stories and let us do our work 😉 #annoying."
Karius is focusing on ousting Simon Mignolet to gain the number one role at Anfield.