Boro have taken just three of the last 24 Premier League points they have contested and slipped to 17th place in the table after Sunday's 1-1 draw at Bolton.
Barnwell said: "When Steve Gibson was appointing Gareth, he put a very strong case for the appointment because he had looked at various domestic mangers who did not satisfy his needs, and he had also considered a foreign manager.
"He felt he had a lot of gifted young players on whom they had spent a lot of time and an enormous amount of money to educate them in the Middlesbrough pattern of life.
"He put a very strong case for appointing Gareth as his manager, and having done that, it is no surprise to us that he will not be swayed by the press or public opinion as to his determination to improve Middlesbrough Football Club.
"Whether that be with Gareth Southgate or anybody else, he has a clear vision of how that club should be run.
"During this period of the season when the culling of managers seems to increase considerably, it is extremely refreshing and a reminder to other clubs that you never achieve your aims if you are continually changing the manager."