McTominay's future has been the subject of speculation throughout the summer, with Man United said to be willing to let him leave for the right price.
Fulham are thought to have had two separate bids for the Red Devils academy product turned down during the summer market, while Napoli have recently been credited with an interest.
According to transfer expert Fabrizio Romano, Napoli have submitted a fresh €25m (£21.2m) offer for the 27-year-old, but Man United have turned down it down, with the 20-time English champions wanting a fee in excess of €30m (£25.5m) for his services.
It now remains to be seen whether the Italian giants return with a fresh new bid in the final week of the transfer market.
Napoli 'fail' with offer for Man United's McTominay
Napoli sporting director Giovanni Manna suggested last weekend that his club wanted to sign McTominay in the closing stages of the summer market.
The Italian giants have also been heavily linked with Brighton & Hove Albion's Billy Gilmour, with a move for the midfielder thought to be close.
Once Napoli have secured an £18m deal for Gilmour, it is believed that the Serie A team could step up their attempts to bring in McTominay.
"They [McTominay and Brighton's Gilmour] are good, strong players. Let's see what happens in the next few days. It's useless to make forward momentum today, there is the game and after the game, we will think about the market," Manna told reporters.
Are Man United signing Ugarte?
Man United are believed to be closing in on the signing of Manuel Ugarte from Paris Saint-Germain, with an announcement expected in the near future.
Following weeks of negotiations, it is thought that Ugarte will move to Old Trafford on loan for the 2024-25 campaign with an obligation to buy next summer.
As a result, McTominay could find it even more difficult to secure regular starts during the 2024-25 campaign, with the Scot already finding himself on the bench for his side's opening two games of the current season.
The midfielder scored 10 goals and registered three assists in 43 appearances for Man United last season, including seven Premier League goals, but it seems that he could leave Old Trafford in the latter stages of the market if the club's valuation is met.