As per The Telegraph, the 20-year-old has managed to impress Marcelo Bielsa this season and the Whites want him to be a part of their first team plans for next season.
The report adds that Leeds can sign the player for £5m.
If they can secure promotion to the Premier League, they will attempt to complete the transfer.
Leeds are looking at the young keeper as a long term investment and it would be wise of them to secure his services.
Meslier has shown that he is ready to play at a high level every week and he could be Leeds’ number one keeper for years to come.