It came after Spurs lost 2-1 at Wembley in a match in which Kane was particularly quiet. The message asked United defender Chris Smalling, "What's in your pocket?" and included a short video clip of him saying, "Harry Kane".
"We have written to both clubs to apologise for any offence caused", an FA spokesperson told Press Association Sport.
The tweet has also been removed.
LONDON, ENGLAND - APRIL 21: General view as Manchester United celebrate after Alexis Sanchez scores his sides first goal during The Emirates FA Cup Semi Final between Manchester United and Tottenham Hotspur at Wembley Stadium on April 21, 2018 in London, England.
Chris Smalling admits he has not spoken to Gareth Southgate about being selected in his England squad for the World Cup, but says he will let his football do the talking. At the end of the day, I've still scored two in my last four, it's not the worst in the world.
"I think I've been a mainstay in the set-up for quite a while now and I just want to finish the season as strongly as I can and see what happens".