The Red Dragons have been without a boss since Sam Ricketts left to join Shrewsbury Town.
Barrow has been in caretaker charge ever since but stresses that he doesn’t want the job full-time.
Having signed a new long term contract he will work with the new manager wherever he is appointed.
That, the coach says, should be sooner rather than later.
“I think they are getting closer, so I would expect somebody in before next Saturday,” he said.
Barrow may have signed off in style with the new era fast approaching.
Wrexham comfortably beat Boston United 3-0 to advance in the FA Trophy on Saturday.
“It was a good all-round performance from the players,” he said. “It was difficult conditions to play in but the start we made gave us a great platform.”