The Serie A giants have allowed centre-backs Leandro Castan and Ervin Zukanovic to go on loan to Sampdoria and Atlanta respectively, and are keen on bringing in a replacement.
Funes Mori, 25, has reportedly been identified as the ideal replacement and any potential deal will likely see Roma opt for an initial loan deal with an option to buy after the season-long loan deal ends.
The fiery Argentine had a mixed season for the Toffees under Roberto Martinez after ousting John Stones from the starting eleven towards the latter stages of the season.
He made a total of 28 appearances in the Premier League and his combative nature was something to be admired until his reckless challenge on Divock Origi resulted in a ban and the striker being carried off on a stretcher.
But new boss Ronald Koeman wants to keep the former River Plate defender and reports suggest he will not entertain any offers from Roma, especially in the event that fellow centre-back, Stones, departs to Manchester City.
The England international has reportedly made it known that he wishes to leave the Merseyside club to join Pep Guardiola. Reports suggest City are preparing a £50m bid for the former Barnsley man this week.
With Stones' future very much unclear and Phil Jagielka approaching his 34th birthday, Everton
According to the Daily Mail, the Toffees are interested in Sunderland's Lamine Kone, who could be available for a fee in the region of £15m.