The PGMOL referee has been videoed making disparaging remarks about ex-Liverpool boss Jurgen Klopp and the club.
There is no date on the video, but it is said to be at least a few years old, given the nature of the conversations in it.
Halsey told The Sun: "If it is authentic then he's got questions to answer, hasn't he?
“But when I saw that, I have to say to myself, is that really David Coote? Has he really said that? To be fair, I was shocked when that video circulated on social media.
"Can he come back and referee from this? I have my doubts.
"And I'm shocked that whoever's posted this video needs sorting big time, because if that was done privately between two of them, then something's not right, is it? Something's not right at all.
"I'm hoping it's not authentic for David's sake. If it is, then he will need all the support from the PGMOL going forward regarding this matter."