The full-back has moved to the City Ground on an initial six-month deal from Crystal Palace.
Having been snapped up by Palace from Huddersfield Town for £2million last summer, Hunt is still waiting to make his competitive bow for the Premier League side.
An unfortunate injury shortly after his arrival was followed by a loan stint at Barnsley over the second-half of the 2013/14 campaign.
Hunt has once again been allowed to go in search of first team football outside of Selhurst Park and the 23-year-old says he would be happy to stay on at Forest if he enjoys a productive run.
He said: “I have only just arrived, so it is a bit soon to be talking about it, but if Forest did want to keep me for the second half of the season, it would be a conversation that we would have to have with Palace.
“But it is such a big club and I don’t see what would stop me from staying if the opportunity arose.”
Hunt added: “My initial goal is to help Forest get up the table and get some games under my belt at the same time. I don’t believe there are many better teams I could have come to in this league.
“I am a Palace player and I do have a long time left on my contract, but I am at Forest now and I will do everything I can to get the club to where they want to be.
“It is also putting myself in the shop window at the same time, but hopefully I can go back to Palace in January and play.
“If it works out otherwise, we will see what happens in the future.”