The 29-year-old midfielder has made four substitute appearances for the Reds since joining the Championship outfit from Tottenham Hotspur on an initial one-month deal and found the net during Saturday's 4-1 win at Barnsley.
"He has been really good," said O'Driscoll. "Sometimes you are wary about getting a big name from the Premier Division. But he has been an influence on things. We have used his opinion on some things.
"Initially, it was a 'suck it and see' thing for both parties. But we would like to keep him if we can, we will discuss things and see what happens.
"The boy has to want to stay. He wants to get his career back on track. But 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."
Jenas is understood to be willing to remain with Forest and he could be available for next weekend's clash with Millwall.