The 22-year-old had found playing time difficult to come by under Zinedine Zidane at the Bernabeu this season and in January, he joined the Gunners on loan until the end of the 2020/21 campaign.
Odegaard has already made eight appearances across all competitions for Arsenal, including six in the Premier League and two in Europe.
But Odegaard is yet to get off the mark in terms of goals and assists at the Emirates, and Arsenal legend Winterburn admitted he needs to see more from the playmaker.
When asked if he’s done enough, Winterburn told Stadium Astro: ‘At the moment, no he hasn’t. Ian Wright was saying he reminds him of Gascoigne, he doesn’t remind me of Gascoigne.
‘For a young-ish player who hasn’t played a lot of football to come into your club I think at the moment I’m sitting on the fence, but I see the potential there. His link-up play is very good, he wants to get on the ball, he’s a good passer.
‘But if I’m looking at creative midfield players, I’m always looking for a bit more. I want to see a creative midfield player who is capable of scoring goals and I want to see them setting up chances or being involved in the chances created of the team.
‘What I want to see from Odegaard is him getting on the ball all the time, trying to play through the tricky passes that will set up Aubamayeng and Saka or the final pass that goes out to a wide area is whipping the ball in for an opportunity for one of our other players to score.’