Jenas' initial one-month stay at the club where he began his career came to an end following Saturday's 4-1 Championship victory at Barnsley, where the 29-year-old scored Forest's final goal.
The midfielder has made four substitute appearances since returning to the Reds and has been praised by manager Sean O'Driscoll.
"Sometimes as a manager you are wary about taking a big name from the Premier League, which he is, but he's been first class," O'Driscoll told Forest's official website. "His attitude has been great, he's got great experience and we've used his opinion on things because he has an influence on the group.
"He wants to get his career back on track and the initial first month was a suck-it-and-see thing for both parties. We would like to keep him if we can. We will discuss things and see what happens.
"Every time he has come on, he has given us an added dimension. He is not far from getting a start and I am sure he will back himself to hold on to that place when he does."
- Sports & Recreation