The FA said in a statement on Sunday that Lingard had been drafted in to help bolster manager Roy Hodgson's ranks following Friday's 2-0 defeat to Spain.
It is Lingard's first call-up to the senior squad, having previously represented England at under-21 and under-17 level.
The 22-year-old has enjoyed a breakthrough season at Old Trafford, making seven appearances in the last month and scoring his first goal for the club against West Bromwich Albion last weekend.
Carrick has returned to United after being taken off on a stretcher with an ankle injury in the final minute of England's defeat in Spain.
England boss Hodgson said he feared the midfielder had suffered ankle ligament damage.
Vardy missed Friday's match due to an injury he suffered playing for his club last weekend and has also been ruled out of the game against France.
The build-up to Tuesday's match has been overshadowed following the deadly attacks in Paris that left 129 people dead and 352 wounded.
The French football federation (FFF) announced on Saturday that the game would go ahead as planned.
(Reporting by Toby Davis; editing by Patrick Johnston)
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