The Portugal international is under contract until the end of the 2026/27 season, although United have the option to extend his current deal by another 12 months, taking it through to June 2028.
United are keen to secure Fernandes on a new agreement rather than rely on that extension, with negotiations between the club and his representatives having been ongoing since September.
However, according to BBC Sport and other reports, the 32-year-old has turned down the latest proposal after also rejecting an initial offer during the European summer.
United are determined to avoid losing their captain without receiving a transfer fee and are prepared to activate the extension clause if an agreement cannot be reached.
The club are expected to exercise that option before the end of 2026, which would prevent Fernandes from entering the final six months of his contract and negotiating a potential free transfer with a foreign club from January 2027.