Rovers, who released a statement of Tuesday evening to confirm the news, moved to wish the 29-year-old midfielder well in his future playing career.
The statement read: "Doncaster Rovers would like to thank Richie for his tremendous service over the past two years, which has helped bring the Club great success.
"Though we are sorry to lose Richie, we understand that at the age of 29, the player has to capitalise on the remaining years of his career, with this possibly being his last chance to ensure security for himself and his family.
"With Richie being 30 years of age in March next season, he made it clear that Leicester City had made him a offer he could not turn down.
"Richie did state that it would be a big wrench to leave Rovers, considering the success he has had at the club."