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Olsen has played three times for Everton - twice in the Premier League in a defeat at Newcastle and an away victory against Leicester, as well as their Carabao Cup defeat at the hands of Manchester United.
His debut against Newcastle ended a run of 120 consecutive Premier League starts for Pickford, whose form has been questioned in recent months despite him currently maintaining his status as first-choice goalkeeper for both club and country.
Ancelotti has now confirmed he wants to sign Olsen on a permanent transfer, in order to have two experienced internationals competing for the goalkeeping gloves.
He told a press conference ahead of Everton's Premier League clash with West Ham United on New Year's Day: “Yes, I think we are pleased to have him Olsen here.
“He is a goalkeeper with experience, so I think we are going to talk with him and his agent to find a solution for him to be here permanently. It can be for January or in June.
“The games that he played he did well. He was comfortable, he was calm, he was quiet. I think he gives good support to Jordan also.”
Olsen, like Pickford, is first choice for his country and has made 42 senior appearances for Sweden.
He and Pickford faced off at the 2018 World Cup, where England beat Sweden 2-0 in the quarter-finals in Russia.
Olsen moved to Roma in 2018 from FC Copenhagen as replacement for Alisson, who had moved that summer to Everton's Merseyside rivals, Liverpool. However, he only served as first choice for the Italian side for a single season before being replaced in the summer of 2019 by new signing Pau Lopez.