The 42-year-old was suspended in November 2024 after the clip showed him making derogatory comments about the German and Liverpool.
Coote was sacked a month later by the Premier League referees’ body Professional Game Match Officials Limited (PGMOL) following a “thorough investigation” into his conduct.
The FA has now charged Coote in relation to rule E3.1, which alleges he acted in an improper manner and/or used abusive and/or insulting words.
It says reference to Klopp’s nationality in the video constituted an “aggravated” breach.
However, Coote has been cleared of any gambling misconduct following claims that the referee had discussed issuing a yellow card to a player prior to a match.
Coote, from Nottinghamshire, had denied the gambling misconduct claims, saying the allegation was “false and defamatory”.