The 23-year-old has been persistently linked with Tottenham Hotspur for more than 18 months, but has so far failed to secure a move away from South America.
Saints are the latest club to shown an interest in the Selecao striker and according to Sky Sport Italia have made a substantial offer to Internactional for his services.
The Porto Alegre-based side have so far stubbornly refused a string of offers for their star player, previously rejecting advances from Spurs in the January transfer window.
Damiao has two years left on his contract at Internacional and insisted earlier this week that he has not received an official offer to leave.
The forward is a regular for his country under Luiz Felipe Scolari, but was forced to withdraw from Brazil's Confederations Cup squad through injury.