Andy Carroll is on the brink of joining West Ham permanently after passing a medical with the Premier League club.
Brighton have threatened to take action against the LMA as the dispute with Gus Poyet rumbles on.
Kane, who scored nine goals in the Championship for Millwall last season during a successful loan spell at The Den, will go straight into the Foxes squad for their game at Blackpool on Saturday.
The England Under-19 international spent the first half of the campaign on loan at Norwich, where he made five appearances before being recalled by Spurs.
Kane has made just one Premier League appearance for Tottenham but has played in the Europa League for the north London club this season.
He has signed a new contract with Spurs to keep him at White Hart Lane until 2017.