David de Gea's contract at Old Trafford expires next week and the club has been linked with Serbian stopper Djordje Petrovic.
However, The Mirror claim De Gea may extend his 12-year stay in Manchester with Ten Hag not willing to spend significant money on a new keeper.
As big money moves are said to be in the works for a new striker and a midfielder, with Moises Caicedo now heavily linked, signing a new keeper may weaken United's ability to strengthen in other areas.
De Gea is currently earning £375,000-per-week at the club, but the Spaniard has been told that he will need to take a significant pay cut if he wants to stay.
The report states that United's latest offer is in the £200,000 bracket, but De Gea's position may be strengthened by the knowledge that Ten Hag wants to improve other areas of the pitch more urgently.
A reluctance to cough up £65m for Mason Mount illustrates United's inability to spend heavily this summer, and could lead to De Gea staying, especially if valuations for the likes of David Raya, Andre Onana and Diogo Costa in particular are out of their range.
De Gea remained a regular for all of the 2022-23 season, but high-profile errors against Sevilla in the Europa League, and in the FA Cup final versus bitter rivals Manchester City led to reports that the former Spanish international would depart.
Despite those notable mistakes though, De Gea still won the Premier League Golden Glove, after keeping more clean sheets than any other goalkeeper in the division last season.
The news comes soon after the Red Devils had been linked with New England Revolution goalkeeper Petrovic, who has impressed in his first year in MLS.
Despite finishing down in 10th in the Eastern Conference, Petrovic still finished in second in the MLS Goalkeeper of the Year vote.
His recent form has also seen him put pressure on Vanja Milinkovic-Savic for a starting spot in the Serbia national side.
However, a report in the Daily Mail claims that Petrovic would not be considered first-choice should United make a move for the 23-year-old.
The article also states that Tom Heaton's future is uncertain, while Jack Butland is on his way to Rangers after his loan spell from Crystal Palace ended, so Petrovic may only have come in as backup.