This comes after Leicester City officials verbally agreed a fee around €10 million, during their trip to Rennes last week.
Sources from both Leicester and France, believe Bourigeaud would be keen on a move, because of the chance to play in the Premier League.
The Frenchman has been offered a new contract by Rennes, but no decision from the player's front has been made, with him interested in a move away from the Ligue 1 side.
Leicester hope to expand on the amount of French talent in their first-team, as Wesley Fofana and Boubakary Soumaré have both played for France at youth level.
Bourigeaud’s strongest position is at central midfield, but he can also operate on both wings. He's a very versatile player, with this being another reason why clubs all over Europe are very interested in him.
Last season he featured 38 times in Ligue 1, scoring 11 goals and registering 12 assists.
He also helped his team reach the UEFA Europa Conference League Round of 16, where Rennes got knocked out by Leicester 3-2 on aggregate.