The 27-year-old Albanian started all three of his country's group games at Euro 2016 last month.
Berisha, 27, joined Lazio from Swedish side Kalmar in September 2013 for a reported €300,000 fee [£251,154] but has since struggled to dislodge regular keeper Federico Marchetti from the line-up, featuring just 11 times in Serie A last season.
The Albania international has fared well when deputising for his Italian counterpart, however, recording 22 clean sheets in 60 total appearances for the Rome-based club, and also played in all eight of his country’s Euro 2016 qualifiers as they secured a spot at the finals for the first time in their history.
He promptly caught the eye of watching scouts with his displays during the tournament's group stages in France last month, and now Villa and Everton are both believed to be interested in luring the 6ft 4in shot-stopper to England this summer.
Villa have been linked with a host of shot-stopping options in recent weeks in the wake of Brad Guzan’s disastrous 2015-16 campaign, with the latest reports suggesting that the club have lodged a £4.1 million bid for Verona’s Pierluigi Gollini [h/t gianlucadimarzio.com].
The Daily Mail also claimed earlier this week that new Everton boss Ronald Koeman is close to completing a £1 million deal for Fulham keeper Maarten Stekelenburg, who spent last season on loan at Southampton, but wants to bring in another potential No. 1 to compete with the Dutchman and Joel Robles following Tim Howard's switch to the Colorado Rapids.
Berisha, meanwhile, has just 12 months remaining on his Lazio contract and now faces an uncertain future at the club with Marcelo Bielsa set to take over as head coach.
Lalaziosiamonoi.it's report states that the 37-times capped Albanian is keen to secure regular first-team football after spending three seasons as a backup and will look to move on this summer unless Marchetti leaves the Stadio Olimpico instead.