The 21-year-old becomes the Clarets' second arrival of the winter window following Chelsea loanee David Datro Fofana and hailed Vincent Kompany for his role in convincing him to make the switch to Turf Moor.
"I feel very good, I'm really happy and ever since I first heard about the move I wanted to move here," Esteve told the official Burnley website.
"I have followed the Premier League since being a young child and had only pictured myself ever being here, so I am really excited. The manager (Vincent Kompany) spoke really positively about the project here and I just wanted to be a part of it straight away."
Esteve strengthens a Burnley backline which has conceded 45 goals in 22 Premier League fixtures this season, the second-worst defensive record in the division behind basement side Sheffield United.
The 2002-born defender has been on the books with Montpellier since 2013 and has now made 62 senior appearances for La Paillade in all tournaments, recording one goal and one assist in the 2023-24 Ligue 1 campaign.
Esteve - who stands at 6ft 4in tall - made his France Under-21 debut in September and will be in line to earn his Burnley baptism when the Clarets host Fulham in Saturday's Premier League clash.
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