The 30-year-old is entering the final year of his existing deal, which carries no option to extend, and currently earns £200,000 a week.
Manchester United are not expected to match that salary for a player who will be 32 before the start of the 2027-28 season and who has missed more than 20 games in six of his 12 seasons at the club.
Despite that, Shaw had one of his best ever campaigns last season, becoming only the fifth player in the Premier League era to start all 38 league games in a single season for United.
Any renewal could mirror the arrangement given to Harry Maguire in April. Maguire's basic salary was reduced from £200,000 a week, but Champions League qualification and additional clauses meant he could earn more overall from the deal, which included a further one-year option.
United are already in the market for a long-term left-back regardless of whether Shaw stays, making his position at the club one of the more complex negotiations of the summer window.
Shaw joined Southampton for £31m in June 2014 and has since made 325 appearances across 12 seasons at Old Trafford, an average of 27 games per season. He is the longest-serving player at the club, having also been a two-time European Championship finalist with England.
The situation reflects a broader moment of transition at United this summer, with the squad being reshaped under manager Michael Carrick as the club attempts to consolidate their Champions League return and compete for major honours next season. Shaw's consistency last campaign gives Carrick reason to want him around, but not at the current wage level.