No, it is not true and there is no evidence for that. The person who has stated it has long claimed that he has the correspondence and has deposited it in a safe box in London, which is ludicrous. The truth of the matter, as explained by Ayatollah's son, was that one of the Iranian students of Einstein, professor Hesabi, meets Ayatollah Boroujerdi in Qum. Knowing that he was a student of Einstein, the Ayatollah asks him about the Einstein's relativity theory. After a detailed explanation and some technical questions by the Ayatollah the latter finally says:
سلام مرا به آقای انیشتین برسان و بگو شما موفق شدید از فیزیک روزنهای به متافیزیک باز کنید
"Convey my regards to Mr Einstein and tell him that you have managed to open a widow from Physics to Metaphysics."
That is all.