Both clubs are expected to be in the market for new central defenders next year, and Le Normand has allegedly been identified as a possible acquisition by the two giants of English football.
According to TEAMtalk, scouts from Man United and Liverpool have returned favourable reports on Le Normand, who started his professional career with Brest before making the move to Real Sociedad.
The six-time Spain international has a contract with his Basque club until June 2026, but it is understood that his release clause is around €50m (£43.6m), which is unlikely to put off interested clubs.
Le Normand made his first-team debut for Real Sociedad in November 2018, and he has since represented the club on 190 occasions, scoring five goals and registering six assists in the process.
The defender has gained Champions League experience during his time at the club, representing them in the competition this season, with the La Liga side top of Group D on seven points.
Le Normand has featured on 12 occasions at club level this season, scoring once, playing in all but one of his side's matches in all competitions during the 2023-24 campaign.
Arsenal were also linked with the centre-back earlier this year, although it is unclear whether the Gunners would be in the running for his signature if it became clear that he was leaving Real Sociedad.
Man United's activity when it comes to a new centre-back next summer is likely to depend on what happens with Harry Maguire between now and the end of the campaign.
Maguire has played his way back into the first XI, helped by another foot injury for Lisandro Martinez, and he scored the winner against Copenhagen in the Champions League on Tuesday night.
The England international also put in a man-of-the-match performance against Sheffield United in the Premier League last Saturday, as the Red Devils recorded a 2-1 win.
West Ham United had a bid accepted for Maguire over the summer, with Man United prepared to let him leave, but the defender ultimately decided against a move.
Jonny Evans only has a contract at Old Trafford until the end of the campaign, while there is uncertainty surrounding the future of Victor Lindelof.
The Sweden international has again been a regular this season, making 12 appearances in all competitions, but his contract is due to expire next summer, and Man United are yet to activate a 12-month extension.