Rob Tanner, author of 5000-1: The Leicester City Story, has claimed on Twitter that Steven Gerrard's side should conclude a deal for the player in the next couple of days.
King was a member of Claudio Ranieri's squad that stunned world football by winning the English Premier League title in 2015-16.
Though he played 29 times in the title-winning campaign he's found regular football difficult to find at the King Power stadium, going out on loan to Swansea City and Derby County in the past two seasons.
If a deal for the 30-year-old midfielder goes through he'll become Rangers' tenth summer signing.
King became the first player in the Premier League era to win the league title in each of the top three tiers in England with the same club after helping Leicester recover from the depths of League One.
He is the club's most-capped player ever with 50 appearances for Wales.