De Gea has started another campaign as the Red Devils' first-choice shot-stopper after fending off competition from Dean Henderson in the 2021-22 season.
Henderson has since joined Premier League newcomers Nottingham Forest on loan, while Lee Grant announced his retirement as a player at the end of the most recent campaign.
Tom Heaton has therefore emerged as Man United's number two behind De Gea, who turns 32 in November and only has one year left to run on his Old Trafford contract.
Man United do possess an option to extend the Spaniard's deal by an additional 12 months, but he was linked with an exit from the club earlier this summer.
Erik ten Hag is therefore said to be actively searching for a new name in between the sticks, and according to The Athletic, Meslier's name has been mentioned.
The 22-year-old displaced Kiko Casilla in the Leeds lineup at the end of the 2019-20 season after arriving from Lorient on loan and is Jesse Marsch's undisputed number one in the Premier League.
Meslier has kept 25 clean sheets in 89 outings for the Whites in all competitions over the past three years and made his move to Elland Road permanent for £5.9m following their promotion to the Premier League.
The France Under-21 international only signed an extension until 2026 last year, though, and the report adds that Leeds will not entertain offers for their number one in the current window.
Following his temporary switch to the City Ground, Henderson blasted Man United for their "criminal" treatment of him, as he did not play a single game in the 2021-22 Premier League season.
"I turned so many good loans down last summer for that reason and they wouldn't let me go. To waste 12 months was criminal really, at my age. I was fuming. It was so frustrating," Henderson told talkSPORT.
"I didn't really want the manager to see me in training because I knew he'd probably want to keep me. I told all the hierarchy: 'I need to go and play football. I don't want to be playing second fiddle.' And it panned out so I was almost gone before the manager came in."
United also have Nathan Bishop and Matej Kovar hoping to force their way into first-team contention, with the former's proposed loan move to Wycombe Wanderers reportedly being put on hold amid the Red Devils' goalkeeper search.
Man United were tipped to move for Watford's Daniel Bachmann at one stage, but the Austrian is no longer expected to leave Vicarage Road in the current transfer window.
Ten Hag's side will be aiming to bounce back from their opening defeat to Brighton & Hove Albion when they travel to Brentford in the second gameweek of the 2022-23 Premier League season on Saturday evening.