The 22-year-old took over from Kiko Casilla as the Whites' number one towards the end of the 2019-20 season and has earned plenty of admirers in the Premier League.
Meslier has kept 27 clean sheets in 103 appearances for Leeds across all competitions so far, having left Lorient for Elland Road on an initial loan deal in 2019.
The Frenchman's move to England was made permanent upon Leeds' promotion to the Premier League in 2020, and he has 18 clean sheets to his name from 87 top-flight outings.
Meslier has only recorded two shut-outs from 14 Premier League games this term and is not due to become a free agent until 2026, but his performances and potential have attracted suitors.
According to Media Foot, both Chelsea and Man United are taking an interest in signing Meslier, but the two clubs are facing stiff competition in the form of Bayern Munich.
The German champions supposedly view Meslier as an ideal long-term replacement for Manuel Neuer should a move for Aston Villa and Argentina number one Emiliano Martinez fail to come to fruition.
However, the report adds that Meslier would prefer to continue in the Premier League should he leave Leeds, handing Man United and Chelsea an early boost in the race for his services.
The Red Devils could potentially wave goodbye to current first-choice goalkeeper David de Gea at the end of the season, as the 31-year-old's contract expires in just over six months' time.
Man United recently activated one-year extensions for Fred, Diogo Dalot, Luke Shaw and Marcus Rashford, but they have not triggered a similar 12-month option in De Gea's deal.
As things stand, De Gea could enter talks with foreign clubs in January, and Atletico Madrid are said to be considering a move for their former player, who left the Spanish capital in 2011 to join Man United.
Meanwhile, Chelsea are also facing uncertainty over both of their shot-stoppers, with Kepa Arrizabalaga emerging from the shadows to relegate Edouard Mendy to the bench in recent weeks.
Unhappy with being demoted to second choice, Mendy is understood to be considering his future at Stamford Bridge and has supposedly rejected the chance to sign a new six-year deal.
Mendy's contract does not expire until 2025 - the same length as Kepa's, whose own long-term career plan is also shrouded in uncertainty despite his recent revival.