The Gunners have already seen three bids knocked back for France winger Lemar, who scored 12 goals in 49 appearances for Ligue 1 champions Monaco last term.
But Wenger is determined to bring Lemar to the Emirates and Arsenal are set to return with a fourth offer for the talented 21-year-old.
The north London outfit have signed Sead Kolasinac and Alexandre Lacazette this summer but Wenger hopes further reinforcements can be secured after an underwhelming campaign.
The Daily Mail say another player attracting interest from Wenger is dynamic Nice star Seri, who is also a long-term target for Tottenham.
It is claimed the Ivory Cost international is on Arsenal’s radar and the club are poised to launch a move for his services later in the summer.
Spurs were believed to be talks with Seri earlier in the window and Nice have confirmed they will have to sell ‘one or two’ players in the coming weeks to fund moves for incoming players.
Nice’s president Jean-Pierre Rivère told L’Equipe last week: ‘There should be three or four more arrivals, and two departures.
‘I think we will manage, because these last ones should be finalised faster than those from last year.
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‘We want to finish these as soon as possible, but certain cases take more time than others.’
Serie A giants Roma have also expressed interest in Seri but are said to be unwilling to meet his asking price – the Ivorian has a release clause of £36million.