The 27-year-old has been a regular for Spurs during the 2019-20 campaign, scoring twice and contributing eight assists in 35 appearances in all competitions.
Aurier has been the club's undisputed first-choice right-back since Jose Mourinho's arrival as head coach, but the Ivory Coast international's future continues to be called into question.
According to Sky Sports News, Monaco could attempt to bring the defender back to Ligue 1 for the 2020-21 campaign.
Aurier won two Ligue 1 titles during his time with Paris Saint-Germain, while he has also turned out for Lens and Toulouse in France's top flight.
Spurs will be looking to keep their slim hopes of claiming a top-four finish in the Premier League table alive when they travel to Sheffield United on Thursday night.