Hartson believes the Arsenal fans would welcome the additions of Everton striker Romelu Lukaku and Southampton central defender Virgil van Dijk.
Both players have been linked with exits, with Lukaku on Chelsea's wishlist and van Dijk rumoured to be wanted by Liverpool, among others.
Wenger is set to sign a new two-year contract at Arsenal, with an announcement on his future set to be made by the club tomorrow.
Hartson doesn't agree with the decision but has demanded big-name signings to challenge for the league title next season.
John Hartson reckons Arsenal should look to sign Romelu Lukaku and Virgil van Dijk Arsenal's £200m summer wish list Wed, April 12, 2017
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Be in for these players so that Arsenal fans can feel that Arsene Wenger has built up a strong enough squad to put on a real challenge
John Hartson on Romelu Lukaku and Virgil van Dijk
"I think there will be protests if he stays, I really do," he said on BBC Radio 5 Live.
"A lot of the supporters want to see a change at Arsenal, that being at the top, the manager.
"What would appease the supporters would be if he brought in some really big names. He makes a statement there by saying 'we've got a strong enough squad'.
"Say for instance Lukaku becomes available at Everton, £60million whatever it is, Van Dijk for instance, £50million. Arsene, the fans are saying 'please be in for these players'.
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John Hartson says Arsene Wenger needs to splash the cash if he stays at Arsenal
"Don't be £5million out or whatever it is, be in for these players so that Arsenal fans can feel that Arsene has built up a strong enough squad to put on a real challenge, a real go.
"He's still got to get (Alexis) Sanchez's contract over the line, no Champions League next season, is he going to attract these big players?
"There's all these things that come into play here."