The Red Devils were favourites in Gdansk but the game went to a penalty shootout after a 1-1 draw over 120 minutes.
After 21 converted penalties, David De Gea missed his spot kick to ensure Villarreal won the trophy for the first time.
McTominay was one of United’s few bright sparks on the night, with the Scot taking the game by the scruff of the neck in the second half.
But it wasn’t enough as an exhausted United sat back in extra-time and lost on the lottery of penalties in the shootout.
‘We will be back,’ wrote McTominay, who accompanied his simple message with a ‘heartbreak’ emoji.
Solskjaer has taken stinging criticism after the defeat, with critics citing the Norwegian’s poor in-game management and failure to rotate for United’s lacklustre display.
United made their first substitution in the 98th minute, while Villarreal made five in normal time.
Solskjaer then stuck with David De Gea for the shootout, despite having a penalty-saving specialist in Dean Henderson on the bench.
MORE : Dean Henderson has ‘not done enough’ to replace David de Gea as Manchester United’s No. 1 goalkeeper, says Rio Ferdinand after Europa League final defeat
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