To be truthful, I had not known how Peres found out.
I do not know if any of the several books written about the matter reveal the identity of the person who told Peres but in yesterday's Israel Hayom carries a comment by Dan Margalit who quotes another report published this past week.
According to Margalit:
Raful, Commander-in-Chief Rafael Eitan, had convened a forum of previous heads of staff in 1981 to inform them of what was about to happen. A member of Chaim Bar-Lev's staff, and Bar-Lev by then was a Member of Knesset and candidate to be Minister of Defense, learned about it and he told Shimon Peres.
In sending his letter to warn Begin not to attack, he caused the operation to be delayed as the concern was that if Peres knows, many others do as well.