The Manchester United forward was criticised for his performance after Saturday's 2-1 defeat to Italy in Manaus, where he started out on the left.
The 28-year-old set up Daniel Sturridge's goal, but missed two chances to score in the second half as England failed to pick up any points.
"How many forwards playing out of position on the left wing could produce a cross like that with their weaker foot?" said Barry on social media app Fanzai.
"I can't believe there's a debate on whether Wayne Rooney should be starting for England. He has to be one of the first names on the sheet.
"I assure you every player in that changing room would want him in the team. If we are going to do something at this tournament which I still think we definitely can we need Wayne Rooney in the side."
The United forward is said to have put in extra training on Monday ahead of Thursday's match.