Manchester United centre-half Keane and Tottenham striker Kane both played parts in helping Nigel Pearson's side sneak into the top six this season.
Keane moved to the King Power Stadium in November and the 20-year-old went on to make 27 league and cup appearances for Leicester after having his initial loan deal extended until the end of the campaign.
Kane was not signed until the end of February. The 19-year-old played 15 times for the Foxes, mainly from the bench, and scored two goals.
Leicester sneaked into the play-offs on the final day of the season courtesy of Anthony Knockaert's injury-time winner at Nottingham Forest, but their promotion dream was ended by Watford on Sunday in the most dramatic fashion.
Knockaert's injury-time penalty, which would have surely sent the Foxes to the final, was saved and it was Watford who sealed a place at Wembley with a goal on the break just 20 seconds later.