The 22-year-old has found playing time hard to come by at Real Madrid this season and wants to leave the club in search of regular football this month.
Los Blancos are willing to allow Odegaard to leave during the current window, but will only sanction a simple loan deal without an option to buy as the club still rate him highly.
Odegaard’s desire to secure regular playing time and Real Madrid’s openness to let him go this month has put several clubs, including Arsenal, on alert.
The Gunners are claimed to have now made an approach for the Norwegian, but it has emerged that Leicester went in for him recently.
The Foxes were interested in the possibility of snapping Odegaard up on a temporary deal when it first became clear that he could be allowed to move away from the Bernabeu.
However, Brendan Rodgers’ side decided against pursuing a deal for the attacking midfielder after preliminary discussions.
It is unclear why Leicester dropped their interest in Odegaard despite holding initial talks.
Odegaard at present is suggested to favour a move back to Real Sociedad.