Coote is one of the Premier League’s most experienced officials and has been refereeing matches in the top flight since 2018.
The FA told the BBC: “These are very serious allegations and we are investigating as a matter of urgency.”
The PGMOL said it takes a “zero-tolerance approach” to any breach of its code of conduct.
“The facts need to be established in light of these very serious allegations,” it added.
“PGMOL board is committed to taking the appropriate action should any breach of that code be proven.
“David Coote remains suspended and subject to an ongoing disciplinary process by PGMOL, separate to the investigation into this matter which will be carried out independently by the FA.”
Leeds said they “are aware of the allegations” and added: “We respect and have full confidence in the FA, EFL and PGMOL regulations and processes. We will be making no further comment at this time.”
Coote was initially suspended by the PGMOL on 11 November pending a full investigation after a video emerged that appears to show him make disparaging remarks about former Liverpool boss Jurgen Klopp. The FA started an investigation on 12 November.
On 13 November, the Sun published photos it says were taken during this summer’s European Championship, alleging that they appear to show Coote sniffing a white powder through a rolled up US bank note.
Coote was working at the Euros in his capacity as a match official.
The PGMOL said it was taking those allegations “very seriously” and launched an investigation.
Uefa, European football’s governing body, has also launched its own investigation and suspended Coote.
Neither the pictures nor the video have been verified independently by the BBC.
Last week, the PGMOL said Coote’s welfare is “important to us” as its investigation into his conduct continues.
Former Premier League referee Mark Halsey told BBC Sport he was “stunned” by the new allegations against Coote and said they would be “damaging for all officials from grassroots all the way up”, but he stressed the importance of waiting for the investigation to run its course.
Halsey added it was also important to look after Coote and believes mental health support needs to be put in place for referees.
He said: “I’m quite thick-skinned, I can quickly move on to the next game, but some can’t and it is something that needs to be put in place, with all the officials at the top level.”