Welbeck will have just one year remaining on his deal with Arsenal at the end of the current campaign when the club will have to take a decision on whether to cash in on the former Manchester United striker or hand him a new deal.
The 27-year-old is not a regular starter with Alexandre Lacazette and former striker Olivier Giroud ahead of him in the first-half of the campaign, and Pierre-Emerick Aubameyang's arrival in January is likely to further hamper his chances of first-team football.
Welbeck has struggled with injuries since his arrival and has also managed just 22 goals in four seasons with the Emirates Stadium outfit. Wenger is not willing to give up on the England international as his versatility allows him to play on the flanks apart from his regular role as a number nine.
However, similar to the stance taken on Jack Wilshere, who was asked to prove his fitness first before being offered a new deal, Wenger wants Welbeck to prove that he can consistently perform on the pitch without spending too much time in the treatment room.
"I want to keep him [Welbeck] because I believe he has gone through a torrid time," Wenger said, as quoted on Sky Sports. "He has come out well. I see signs in training now that are positive. If he can stay on the pitch and not be injured, I want him to stay here."
"I like him because he can play left, right or up front. For me he is a useful player, first of all because of his quality.
"He might not be a goalscorer like Harry Kane for example, but for England he can perfectly play with Harry Kane because he can play on the flanks," the French coach added.
Welbeck will start for Arsenal when they take on Ostersunds in the Europa League last 32 second-leg on Thursday (22 February). He has not scored for Arsenal since scoring the only goal in the 3-1 FA Cup loss against Nottingham Forest at the beginning of January.