An allegation of racism by Stockport player Ryan Johnson was made against a Barnet staff member on Saturday.
The backroom coach in question attended training on Monday after the National League club said police had concluded there was 'no allegation or substance for a criminal investigation to take place'.
But captain Jamie Turley confronted the accused staff member, who then complained, resulting in both being suspended by the non-league club.
Outraged by Turley's suspension, the Barnet players threatened to strike, but have since been talked down.
With the situation still not settled, Kleanthous gave his version of events.
He told White and Jordan on talkSPORT: "The Stockport player was taking a throw-in, standing next to the linesman. He heard a comment which he believed was racist.
"I am certain he did believe that, so there's no question about that.
"The issue is, the person who said it says he said something to the linesman, not the player, and the comment he made was different to the one the player thought he heard.
"There are a number of witnesses to that, who the police have spoken to and corroborated that."
He added: "On Monday morning, all the players weren't aware this was going on and weren't properly made aware.
"Unfortunately, Jamie Turley confronted the individual in a manner which meant the individual had to raise a complaint, which in itself was a very serious complaint.
"So I had no choice in those circumstances. I have to suspend both and let tempers cool and let HR investigate in the correct way, getting both sides of the story."
Kleanthous confirmed the players threatened strike action, but insisted he had to fight back or risk being kicked out of the National League.
He continued: "Because the player has been suspended, I've been told the other players said they didn't want to play if the player was suspended.
"When I was approached by the manager about that, I said, 'Look - you can't have a situation that we're not going to play a football match because someone heard something'. I don't know where that would take us.
"If that happened, the National League may throw us out."
Kleanthous hopes the players see the club doing their best to follow the right procedures to ensure the correct conclusion is reached.
He said: "Barnet does not tolerate any form of racism. Not only racism, but any form of discrimination, abuse or otherwise. We are not that kind of club.
"I think it's very important that the club is educated on the process that where an accusation is made, we have to go through.
"The process is different on a matchday because the safety officer controls the stadium. If it involves racism, it's different again because it goes straight to the police.
"The PFA will be here, I'm hoping Kick It Out will be here, so we can have an open discussion that lets the players know we are doing everything the right way."