Tudgay's existing deal is due to expire next month, when he could leave the City Ground on a free transfer.
The 30-year-old's agent, Darryl Powell, has confirmed that his player is considering Forest's offer.
"Forest have offered Marcus a new one-year deal and he is weighing up all his options," Powell told Sky Sports News.
"Marcus loves it at Forest and playing under Billy Davies, even though he would have liked a bit more security with a longer-term contract offer."
Tudgay enjoyed a successful two-month loan spell at Forest's Championship rivals Barnsley last season, scoring three goals in nine appearances before returning to the City Ground and featuring in two games at the back end of the campaign.