Boro have endured a disappointing start to the season and sit 16th in the Premier League after a run of seven games without a win.
"We have played 12 games in the Premier League, not got the results we have wished for and already there is a noise about the continuation of the manager," Gibson told the Mail on Sunday.
"It is a nonsense. There has even been some trash about us having lined up a replacement. We know that's not true and Gareth knows that's not true.
"I've spoken to Jonathan Woodgate and one or two others to tell them it's c***.
"We gave Gareth a five-year contract because he had to be given a chance to develop. If we go 1-0 down at Bolton I will feel horrendous, but it won't shape what we do at the club."
Gibson also revealed that he considered appointing a German manager before naming Southgate as Steve McClaren's successor in the summer of 2006.
He added: "I looked at a German coach. He wanted to bring in a German fitness coach, German masseur, German players. We were going to become Middlesbrough-on-Rhine.
"It didn't feel right. We had all these kids coming through from our academy. An outside manager might not recognise that, but Gareth knew.
"He is a good man, with integrity and fantastic football experience. He will make mistakes as he's inexperienced, but he has the intellect to learn from those mistakes.
"That is why we went for Gareth over more experienced people."