Correio da Manha says Arsenal directors met with Gyokeres' agent, Hasan Çetinkaya, on the weekend.
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The Sweden international is a target for Arsenal, where manager Mikel Arteta is a big fan.
Arsenal officials informed Cetinkaya a willingness to pay €70m to Sporting for his client, who would receive a signing bonus of €12m and a five-year contract worth €7m-a-season.
Cetinkaya will now take Arsenal's offer to Sporting president Frederico Varandas in the coming days, with a meeting now being scheduled.