Greenwood, 19, was expected to make Gareth Southgate's 26-man party for the European Championships after returning to form at the end of last season.
However United say he will remain in Manchester for treatment on an "underlying injury".
BBC Sport understands Lingard has failed to make Southgate's final squad.
Lingard, 28, excelled during a loan spell at West Ham United in the second half of the Premier League campaign - scoring nine goals and making a further five assists in 16 appearances for the Hammers.
England boss Southgate will name his final squad for the tournament on Tuesday at 17:00 BST, with six players now due to be cut after Greenwood pulled out.
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