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Arsenal scouts were in attendance to watch the 25-year-old play 90 minutes in yesterday’s 4-2 league win over Rio Ave, according to A Bola.
Danilo, who has 17 caps for Portugal, is under contract until 2020 but has a £34.5m release clause.
He featured five times for his country during their shock triumph at Euro 2016.
The Guinea-Bissau-born enforcer has fought his way back to the top of the Portuguese game after being released by Benfica in 2010.
Arsenal boss Arsene Wenger has used a host of combinations in midfield this term.
Aaron Ramsey, Francis Coquelin and Santi Cazorla have all spent time on the sidelines, Granit Xhaka has been sent off twice while Jack Wilshere is on loan at Bournemouth.