Lyon announced on Tuesday that they had rejected a £29.5m (€35m) bid from the Gunners for Lacazette, although Squawka has learned that the offer was made back in June.
Arsene Wenger has been assessing his options to improve his strikeforce in the summer transfer window, having missed out on Jamie Vardy after the 29-year-old rejected a £20m switch to north London to stay at Leicester.
Lacazette, 25, is believed to be considered the most attainable target for the Gunners in an inflated market in which Juventus paid £78m for Gonzalo Higuain and Romelu Lukaku would cost more than £60m.
But Wenger does not want to pay more than £33.7m (€40m), the same sum that Chelsea spent to sign Michy Batshuayi from Marseille.
Wenger believes the two players are of a similar standard having scouted both throughout last season and does not want to break the bank for Lacazette.
There are some concerns that Lacazette may not suit Arsenal’s style of play and the Londoners are ready to walk away should Lyon reject an improved offer of £33.7m.
Lyon described Lacazette as “irreplaceable” in their statement on Tuesday while the club’s president, Jean-Michel Aulas, has said “it would be malpractice” to sell their star striker.
Lacazette, who scored 21 league goals in 34 appearances last season, is under contract until 2019 but Wenger feels he could provide strong competition for fellow Frenchman Olivier Giroud.
Atletico Madrid have also been linked with Lacazette while West Ham had a £31m bid rejected earlier in the summer before being told the player had no interest in a move to east London.
Meanwhile, Wenger expressed his disapproval of Lyon’s decision to make Arsenal’s bid public during the week.
“Lyon is responsible for their own statements,” he told the Arsenal club website on Wednesday.
“On the transfer front, I think the best [option] is always to keep as secret as possible. If we sign somebody, we will announce it. But I cannot tell you much more on that.”