Bannan joined the Rams on a month-long loan from Aston Villa on Friday and made his debut the following day as a second-half substitute in the 4-2 defeat to Sheffield United, scoring his first-ever senior goal at Bramall Lane.
The 19-year-old's only previous first-team action was two appearances for Villa in this season's UEFA Cup.
"I think he will be playing a major part in the next few weeks," said Clough. "He is a technically good player and although small in stature, he has a big heart.
"His attitude is brilliant and you saw at Sheffield United in the first five or 10 minutes he was on, he won two or three headers. Now when you are 5ft 5in, to go and win headers in midfield shows great character.
"He is not frightened and has got a lovely left foot on him. We are delighted he is with us and we are looking forward to him being around for the next few weeks."