In the letter, East Antrim MP Mr Wilson said he was “astounded and appalled” by a response from Mr Poots, and questioned why a “unionist minister” was not resisting departmental officials, the BBC reported.
The BBC said the correspondence was made available as a result of a Freedom of Information request.
It related to post-Brexit checks on pets travelling from Great Britain to Northern Ireland.
Mr Wilson accused the officials of being “more intent on implementing the protocol” than recognising the “grave harm” it caused.
He also challenged Mr Poots for “permitting discussions” within his department, which were at odds with the DUP position, the report added.
The letter was sent to Mr Poots in May, just three days before the Lagan Valley MLA was elected DUP leader, the BBC said.
When contacted directly by the News Letter, Mr Wilson said it was a “private letter” and therefore he would not be commenting.
Mr Poots was replaced as party leader by Sir Jeffrey Donaldson last month.
The letter states: “I am surprised that you are permitting discussions to go on within your department to look at how this part of the protocol could be implemented when your party’s position is that there should be no implementation of the protocol and indeed it should be replaced.
“I trust that you will reconsider the answer which you have given to me and give some hope to those who are looking for political leadership to destroy this protocol rather than meekly bow to the unreasonable demands which it makes.”
Asked for his response to the letter, Sir Jeffrey said “efforts should be united” against the protocol.
“The issue raised in relation to pet travel was resolved,” the BBC reported Mr Wilson saying.
“This is one of many issues that have been challenged by DAERA and other departments in Northern Ireland.
“Significant evidence about the unsustainable and unworkable nature of the protocol has been highlighted to HM Government as a result of such cases.
“The recent command paper is a welcome first step by the [UK] government which has demonstrated that the problems of the protocol are now recognised.
“There now needs to be action to remove the border in the Irish Sea.
“Efforts should be united now in pressing both HM Government and the European Union to undo the damage caused by the protocol.”