The paper say that Foxes boss Brendan Rodgers has sanctioned the move, with no room for the English Premier League winner in his current squad.
It's said that he could even be included in Rangers' squad against East Fife in the Scottish League Cup on Sunday, becoming the club's tenth summer signing.
The financials of the deal have not been reported but the player earns £40,000-a-week according to Sportrac, so it's understandable that Leicester would be looking to get at least some of that off their payroll.
Steven Gerrard already has a star studded midfield at Ibrox and adding another experienced campaigner with trophy winning experience will certainly strengthen it.
It may be bad news for one player in Glasgow though - Greg Docherty.
The 22-year-old is currently on the fringes of Gerrard's first-team and the addition of King would surely push him further down the pecking order and potentially out of the running for selection completely.
Steven Davis, Glen Kamara, Joe Aribo, Scott Arfield and Ryan Jack are all picking up more appearances than the former Hamilton Accies man.
He's managed to make a start against St Joseph's, as well as coming off the bench against Progres and Midtjylland. However, he's not featured in either of the side's Scottish Premiership matches.
Docherty was sent out on loan last summer to Shrewsbury Town and excelled, winning player of the year awards after 50 appearances and directly contributing to 21 goals (Transfermarkt).
He's stated his desire to make it at Ibrox as a supporter of the Gers (BBC Scotland), but the addition of King could be a bitter blow to those chances.
With the transfer window still open in Scotland, England below Championship level and abroad, it's even an arrival that could spell the end for player at Ibrox as early as this summer, with time left for any interested parties to make their move.
What more can he do to prove to Gerrard he should be a go-to choice in the middle of the park? It may be time to move on the for the benefit of his career.
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