The midfielder has struggled to make an impact for the Premier League side this season.
The Star reports that the Owls are monitoring both Gibson and his Goodison Park teammate Aiden McGeady, who have both been told that they can move elsewhere on loan during this window.
However, Wednesday will have to see off competition from both Aston Villa and Sunderland, who are hoping that the lure of remaining in the Premier League will be enough to convince the former Manchester United man - who has made just four substitute appearances for the Toffees in the league this term - to join them as they both battle to beat the drop (via Star).
While joining Carlos Carvalhal's side would force the Republic of Ireland international to drop down a division, the positivity surrounding Hillsborough currently may well do enough to convince him to ignore the top flight interest.
Wednesday find themselves currently occupying one of the coveted play-off places in the Championship. And if their recent run is anything to go by, the chasing pack will do very well to catch them.
The club have lost just once in their last eight games in all competitions, and Carvalhal will know that adding a player of Gibson's experience will improve their chances of promotion immensely.