Tom Morley said he employed the company when he moved to Melbourne this month and was "ignored and hung up on" by customer service staff when he called to report it had been two weeks and his packages still had not arrived, The (Brisbane) Courier-Mail reported Tuesday.
Morley said he sent an email directly to the head of the company, multimillionaire Colin McDowell, saying he would contact fair trading authorities if the situation was not resolved.
McDowell replied Monday, blaming Morley for the problems with the delivery.
"We don't want you threatening us," the email states. "Did you secure the connote (consignment note) to the item? If you did not and it came off, how do we track and find it ... and then you blame us."
A second email soon arrived saying the packages had been found in Melbourne without a consignment note.
"So the problem was always yours, we expect an apology but undoubtedly won't get it," the email said. "Maybe we will report to our legal advisers for threatening behavior."
Allied Express declined to comment, the Courier-Mail said.