Have you ever turned down your man again and again, hoping he would go somewhere else? What might be some things you would do to end the relationship and make it look like it was him doing it? What might you do to push him away?
Why do men think that women use sex as bargaining power?
I am not suggesting that across the board, women don't do it, because some do.
Did you ever consider perhaps that if a lady is unhappy in a relationship, that it would more than likely translate into the bedroom?
If I was unhappy, I wouldn't be encouraging you to go have sex elsewhere, I would be really concerned about the emotional state of play.
Communication is key.
The way you're describing suggests that the lady is lying and manipulating his perspective in order to end the relationship, which is just wrong and disrespectful. Like BiAussieGirl said, communication is key. Instead of stringing the person along and hurting their feelings when they care about you, just discuss it with them and tell them that the relationship isn't working. You'll leave knowing that your partner is aware of what genuine reasons you had that convinced you to consider ending it.
If it's his fault, I'd tell him.
But if it wasn't, I would NOT convince him to think otherwise...