The 22-year-old did have a £21.9m (€25m) release clause in his contract, which has another four years to run, but Arsenal offered the extra €5m to beat off competition from Napoli.
The Gunners have recently been heavily linked with the Uruguayan, who is set to showcase his talents at the World Cup later this month.
He joined Sampdoria in the summer of 2015 from fellow Italian side Pescara and has gone on to make 74 appearances.
New boss Unai Emery has made signing a defensive midfielder a priority this summer as he looks to strengthen the squad he inherited from Arsene Wenger that finished sixth in the Premier League table last season.
The Gunners have signed Swiss defender Stephan Lichtsteiner on a free from Juventus, while a move for Borussia Dortmund's Greece centre-back Sokratis Papastathopoulos will be confirmed in July.
It is understood that there is ongoing contact between the clubs and that the deal will hinge on the structure of the payments, with Arsenal ready to spend in excess of £25 million if the fee is spread over several years.