Bilic revealed that Payet had been banished from first-team training at his press conference on Thursday.
The midfielder is refusing to play and wants to return to France.
Bilic wrote in his Evening Standard column: "This situation with Dimitri has been developing for a little while now. Two days ago, he and I had a meeting. I won't disclose what was said, but I was left in no doubt as to his intentions.
"Yesterday, just before the normal pre-match press conference, I asked Dimitri one more time: 'Are you standing by what you said a couple of days ago or have you thought more about it?'
"He replied that he was standing by his decision, so that was that. The right thing to do then was to tell the supporters and the players what was happening and I informed the squad later that morning, before we began training."
Bilic admits that he would love to have the France international available but must now plan for life without him.
"It has left me sad, angry, but at the moment Dimitri has said he will not play for the club, so I have no alternative but to plan for Saturday's match without him," he said.
"He will not train with the rest of the squad and the club's position remains consistent, which is that he is not for sale."