Chelsea alleged that Clattenburg used inappropriate language towards their midfielder John Obi Mikel.
It is understood England defender Cole - a witness for John Terry in his court case following the accusations made by Anton Ferdinand - was one of the Chelsea players interviewed by the FA.
Clattenburg will miss a fourth weekend of matches as Chelsea's complaint is investigated - with a decision from the FA expected later this week.
The referee does not want to take charge of any matches while the case is ongoing and so was not considered for any of this weekend's games.
The FA have interviewed several Chelsea players, including Mikel, as well as Clattenburg, assistant referees Michael McDonough and Simon Long and fourth official Michael Jones.
An independent QC is studying all the evidence in close detail before recommending a decision.