The Argentina defender has been a big hit since arriving at Goodison Park last summer.
The Italian side are desperate to reinforce their defence this summer after allowing Leandro Castan and Erkin Zukanovic to depart.
That has led them to looking at Funes Mori, who they believe would add some much needed steel to their back line.
He only arrived at Goodison Park last summer from River Plate, but quickly established himself as a fan favourite with his willingness to stand up to the physical side of the Premier League.
His performances appear to have impressed Roma, who are now ready to make a bid for him.
According to Gazzetta dello Sport, they are preparing an initial loan with an option to buy the Argentina international should he prove a success.
That is likely to be rejected though, with the Italian newspaper reporting that Ronald Koeman is keen to keep Funes Mori on Merseyside.
He looks likely to lose John Stones in a big money move to Manchester City this summer and so cannot afford to lose Funes Mori at the same time.