The tweet received thousands of likes and retweets but has since been deleted.
"We have written to both clubs to apologise for any offence caused," an FA spokesperson told British media.
Kane, who has scored 37 goals in 44 appearances across all competitions for Spurs this season, struggled to impose himself as Tottenham lost 2-1 to United at Wembley.
United will face Chelsea, who beat Southampton 2-0 on Sunday, in the FA Cup final on May 19.
(Reporting by Aditi Prakash in Bengaluru; editing by Amlan Chakraborty)
