According to a report from Il Romanista, new Roma manager Jose Mourinho wants to offload the Swedish goalkeeper in the summer, even being set to offer him at a bargain price.
Olsen has impressed Carlo Ancelotti while on loan at Everton, and even though he has a contract at the Stadio Olimpico that lasts until 2023, he will most probably not feature again for the Italian side.
Olsen has also spoken of his desire to stay at Goodison Park beyond this season on the club's official website.
He said "We want to stay, but it is not me who decides".
"I can only cross my fingers and see what the club says. As a family, we have moved around and feel we are in a good place now."
"We are happy, so why not stay until it is time to move home to Sweden? That is what we could choose."
Olsen joined Roma in 2018 from FC Copenhagen on the back of an impressive World Cup campaign with Sweden.
While he was the starting goalkeeper for a large part of his first season at Roma, he was then replaced by Pau Lopez, who joined from Real Betis for €23.5 million, leading to Olsen being loaned out to fellow Serie A side Cagliari, and then to Everton.
The 31-year-old has been a fairly good addition to the Everton squad, mainly being a backup goalkeeper for Jordan Pickford, offering another option between the sticks.
With the arrival of Mourinho at Roma, Olsen's future at Roma seems to be in jeopardy even more, especially since the Portuguese manager is targeting plenty of changes in the whole squad, with reports claiming that Manchester United goalkeeper David de Gea is on his list for transfers this summer.
Moving to Everton seems like the best option for Olsen, who has done very well to keep Pickford on his toes, and with him having the approval of Ancelotti, signing him would be very good business by the Toffees, especially for the bargain price of just £4.5 million.