Hull are hoping to bring in Elabdellaoui as the club's only recognised right-back, Ahmed Elmohamady, is away on international duty with Egypt at the African Cup of Nations.
Elabdellaoui started his career at Manchester City but failed to make the grade at the club and after several loan spells away from the club, he has ended up at Greek champions Olympiakos.
Elabdellaoui worked with current Hull manager Marco Silva last season, helping the Greek giants to their sixth successive Greek title.
Silva is hoping that to conclude a deal for the Norway international before Sunday's showdown with Chelsea at Stamford Bridge.
Should the Hull boss fail to sign the 25-year-old, Harry Maguire is likely to continue in the unfamiliar role of right-back.