Reports in Italy suggest that the Biancocelesti are ready to sell the 27-year-old in order to raise funds for Galatasaray striker Burak Yilmaz.
Ederson only moved to the Stadio Olimpico last summer after spending seven years in France with Nice and then Lyon.
However, he struggled to dislodge fellow Brazilian Hernanes and also Antonio Candreva in the Lazio midfield for much of the campaign.
As a result, boss Vladimir Petkovic may be prepared to let the player go in order to raise vital transfer funds.
The report states that Everton and Sunderland have showed tentative interest in the €5million-rated player this summer but could now step up their efforts to sign him now Lazio appear willing to sell.
Toffees boss Roberto Martinez is keen to add to his squad before the transfer deadline though much will depend on the futures of Manchester-targets Marouane Fellaini and Leighton Baines.