Arsenal have made regular checks on Claude-Maurice, who has scored 14 goals in the second tier of French football this season.
The 20-year-old, who has been capped at under-20 level by France, is one of the hottest properties outside Ligue 1 and is a former team-mate of Arsenal ace Matteo Guendouzi.
Claude-Maurice is rated at around £9million and fits into the budgets of both the Premier League sides now that the transfer window has opened.
But they are facing competition from France where Lille want him to replace Nicolas Pepe who has been watched by Arsenal, Manchester United and Liverpool.
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Arsenal boss Unai Emery is one game away from guiding the club back into the Champions League (Image: GETTY)
Arsenal want players with 'experience and youth' says insider
"We need a whole new backline,” the Scot boldly stated of the ninth best defence in the league on Sky Sports.
“I am disappointed that Petr Cech is retiring. I would have encouraged them to give him another year.
“Bernd Leno has got mistakes in him and would have benefited from Cech giving him a helping hand.
"We are looking for four defenders, so on top of the crazy situation of letting Aaron Ramey go for nothing, it is not great. They need two centre-backs and two full-backs.”
Arsenal face a struggle in keeping Lacazette from joining Barcelona (Image: GETTY)