Boss Wenger has penned a contract extension at the Emirates until 2019 to end months of speculation over his future
The Gunners won the FA Cup after beating Chelsea 2-1 on Saturday and Wenger's new contract was ratified by the club's board this week
But they finished fifth, 18 points behind the Blues in the Premier League, last season to miss out on the Champions League for the first time under Wenger.
And Winterburn, who won the Premier League in 1998 with Arsenal under Wenger, insists it is time for the Frenchman to end Arsenal's 13-year title drought.
"They've made the decision, so it's now time to put in a serious title challenge," he told Sky Sports News
"There's probably 20-25 per cent that would like to have seen a change.
"He's in charge and we're now expecting some significant signings and I would love to see the team progress
He's come through a difficult period
They went backwards after going into the new stadium - now's the time to be serious for the title
"I know a lot of people have been saying typical Arsenal that they make this announcement at the end of the season
It's my understanding there was no decision until after the FA Cup final.
"The club statement was positive and it will be interesting to see what happens in the transfer window."
Former Gunners goalkeeper Bob Wilson warned Arsenal and Wenger they may struggle to attract top signings having failed to reach the Champions League.
"How are you going to get Antoine Greizmann to sign? Without the Champions League, the only way is if you can compete salary-wise to attract the players,' he told BBC Sport.
"Chelsea bought the best player available last season in N'Golo Kante and Arsenal went for him too but came nowhere near the salary that Chelsea pay.
"So the board has to be realistic and say, 'Are we going to compete at this level?'"
Source : PA