The midfielder's one-month loan deal with Forest ended after Saturday's 4-1 Championship win over Barnsley, where Jenas scored their final goal.
Forest manager Sean O'Driscoll hopes to prolong the 29-year-old's stay after making a positive impact at the club.
"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."