It had also been reported another of the party’s nine MLAs, Alan Chambers, had been told not to put his name forward when his North Down constituency association meets to select their candidate.
“I’ve said to Alan you’re as entitled as anyone else to apply for the job for reselection,” Burrows said.
He added the “process should be fair, open and transparent, and if anyone else wants to apply for that job, that is their right under our rules”.
Burrows also said he was unaware of “any risk to Andy Allen’s seat” in relation to the party’s East Belfast MLA.
“All my MLAs are valued. They have the opportunity to go for reselection, I hope they do and I want to grow the Ulster Unionist Party,” he added.
Burrows denied being “on the back foot” as a result of internal party issues, saying that he was “on the front foot”.
“There is endless speculation in the media about personality issues,” he said.
“I think it is pretty poor that is the amount of focus that we are now having, whenever actually we have an opportunity to talk about what in Northern Ireland needs to happen,” he said.
“That’s why there is disenchantment and people want change. I’ve come into politics to offer change.”