The 22-year-old has won rave reviews since joining the Serie A club from FC Twente two summers ago – and, rated at £25million, he has also attracted interest from Manchester United.
Andersen is on the brink of Denmark's senior side after winning 14 Under-21 caps.
And reports in Sampdoria's home city Genoa claim two key club officials recently travelled to London for talks with the Gunners and Spurs.
Sporting director Carlo Osti and lawyer Antonio Romeo, the advisor to club president Massimo Ferraro, are said to have flown to the capital.
Arsenal raided Sampdoria in summer to land midfielder Lucas Torreira in a £26m deal.
And boss Unai Emery could now swell the hard-up Italian club's coffers once more by landing Andersen when the transfer window opens in January.
But the Spaniard may face competition from Spurs chief Mauricio Pochettino, who risks losing key centre-backs Jan Vertonghen and Toby Alderweireld when their contracts expire next summer.
Sampdoria paid Twente just £1.25m for Andersen in 2017, but have seen the Dane's value soar with a series of commanding displays.
The defender previously spent four seasons with the Eredivisie club after leaving home-country club Midtjylland.
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