Off contract in June, Lingard has resisted United attempts to open new contract talks.
The Times says despite Lingard's playing prospects looking slim, United warned six top-flight clubs off making a move for the attacking midfielder, according to The Times.
The publication reports that he is ready to run down the final year of his contract unless it is made clear to him that he has a first-team future with Solskjaer's side.
However, suitors were told Lingard was not for sale when they enquired about him during the summer transfer window.
It is claimed that Lingard hoped that United's refusal to sell him meant he would be set for more playing time following his blistering form on loan at West Ham during the second half of last season.
But it hasn't worked out that way so far, with Lingard relying on this week's chance with England for his first competitive start of the season.