Gardner has joined Lincoln on-loan to the end of the season.
The youngster had signed a new contract with Forest ahead of making the move.
Gardner said of his new deal: “I'm delighted to extend my stay at the Club. I've been here a long time, and it means a lot to sign a new contract and continue my journey here. I'd like to thank all the coaches who have helped me over the years, I'm really excited for the future.
“Going on loan is a good opportunity for me, and I'm looking forward to the challenge of playing regular senior football at Lincoln. I want to test myself, learn, and make an impact on the pitch."
Craig Mulholland, Head of Football Development & Talent Management, also told Forest's website: "Today is a special day for Joe and his family, signing an extension with the club he supports and undertaking his first loan in men’s football with a very progressive club in Lincoln.
“We believe in Joe’s potential and this well-deserved deal is testimony to the dedication he has shown in driving his own development, expertly supported by many outstanding staff here at Forest.
“He has hit many of the key development markers our player pathways team use data to monitor, including a first team debut under Nuno, regular exposure to first team training, men’s 1st team football in the EFL Trophy, cross border competition in the PL International Cup, regular minutes in PL2 and now an important loan experience, at the right time in this carefully planned journey.
“Our excellent staff in the player pathways space will now closely monitor and support Joe in the next step in his journey.
“Everyone at Forest would like to congratulate Joe on his new deal, thank Lincoln and all of the other clubs who tried to secure Joe for their presentations, and of course wish Joe and Lincoln good luck for the second half of the season.”