All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
After careful research, we could not find any basis for this report. It seems that it is one of the stories that are related in sermons and speeches without any chain of narration. However, among what historians recounted about Mu’aawiyah is what Ibn Katheer mentioned in Al-Bidaayah wa An-Nihaayah:
"The Byzantine Emperor had great hopes of attacking and defeating Mu’aawiyah after the latter had instilled fear in him and humiliated him and vanquished his troops. When the Byzantine Emperor saw that Mu’aawiyah was preoccupied with fighting ‘Ali, he approached some cities (on the border) with a huge army, filled with hope of victory. So Mu’aawiyah wrote to him saying, 'By Allah, if you do not desist and go back to your land, O cursed one, I shall reconcile with my cousin and we will unite against you, and I shall certainly expel you from all of your land! I shall keep pursuing you, and the earth, vast as it is, will be straitened for you.' At that, the Byzantine Emperor was scared off and abandoned his plans; he sent word seeking a peace treaty."
Allah knows best.