The France international, who has started all of Arsenal's Premier League matches this season, struggled to influence during the Gunners' 1-0 defeat to Manchester United at Old Trafford on Sunday,
Winterburn highlighted that Wenger's attacking options will be bolstered when Lukas Podolski and Theo Walcott return from injury but feels it is important for the club to sign another striker who offers similar attributes to Giroud.
"One key area which I think everyone is debating is the centre forward role," the 49-year-old told Goal. "Giroud is having to do a lot of work, without a rest, so it's hard work for him.
"I would like to think that they may try and bring someone in, whether that will be a permanent signing or a loan signing.
"If you get the likes of Walcott back, Podolski back, then the bench is looking completely different. You've got a lot more midfield and attacking options. I don't know Wenger's thinking about [them] coming back, and if they can fill that role, but they are completely different to Giroud.
"The way Arsenal play you need that good link-up striker. I just feel that a centre forward would be the main objective for Arsenal, to cover Giroud if he loses a bit of form or gets fatigued. I think that's very important."
Following Sunday's loss to the Premier League champions, Winterburn admits his former side missed out on a great chance to extend their title push but believes the club can challenge this season.
"I think [Arsenal] will be very disappointed," he continued. "I think it was a fabulous opportunity for them. But then you can role reverse that and say it's not ben as costly as it could have been.
"The first half performance was very flat. United looked a little bit more eager in terms of their willingness to work but I didn't think the first half was that great for either team.
"Then the set play just undid Arsenal. I think overall it shows there's not a great deal between the two teams.
"Obviously Arsenal would have liked to come away with at least a point, but I think when they reflect and look at it they will still be pretty satisfied with the way they have started the season.
"And if Manchester United are to be one of the title contenders then Arsene Wenger will believe his team should be around there, if you judge the teams on the two performances today."
When asked if Arsenal's season may dip following their second Premier League loss of the season, Winterburn replied: "I don't think so. They were on a good run last year and then lost to Villa, but they came back very strongly.
"The only thing you can say is that becuase Arsenal haven't been in a [title] challenging position for a while, people will say 'we don't know whats going to happen with them'. Thats why there's doubt there.
"The signs I've seen so far this season have been very good and make me think they are going to be up and around the top. If Arsenal don't win their next game then people will start to doubt them. But at the moment I don't think there's any reason to do that."
