Keane is said to be a target for Israel officials, who are ready to replace Elisha Levy as national boss with his side facing an almost impossible task of making next summer's World Cup in Russia.
Levy, who assumed control in 2016, has games to go in qualifying Group G against Macedonia (H), Italy (A), Spain (H) and Liechtenstein (A) but they currently sit in fourth spot, five points adrift of the joint-leaders, Spain and Italy.
Former Sunderland and Ipswich boss Keane was believed to be among those being considered for the post should Levy go but no approach has been made to the Irish FA for the 46-year-old.
Keane is believed to be happy and settled with his role working under Martin O'Neill as the duo try to help guide Ireland to their second-successive major tournament but is also thought to have designs to get back into management himself at some point.
Ireland are well placed to make Russia in 2018, as they are second in Group D, level on points with leaders Serbia after six games played.