The closest historical account (Prosper, mid-450's) doesn't mention any particular dialogue. The common quotation attributed to the Pope comes from a later text, which presented a fairly mythologized account. The text from Prosper (translation from Readings in European History - J. H. Robinson)

"Now Attila, having once more collected his forces which had been scattered in Gaul [at the battle of Chalons], took his way through Pannonia into Italy. . . To the emperor and the senate and Roman people none of all the proposed plans to oppose the enemy seemed so practicable as to send legates to the most savage king and beg for peace. Our most blessed Pope Leo -trusting in the help of God, who never fails the righteous in their trials - undertook the task, accompanied by Avienus, a man of consular rank, and the prefect Trygetius. And the outcome was what his faith had foreseen; for when the king had received the embassy, he was so impressed by the presence of the high priest that he ordered his army to give up warfare and, after he had promised peace, he departed beyond the Danube."