The Toffees raided Roma during the last transfer window to add Olsen to the ranks at Goodison Park and they have been pleased with what they have seen of the shot-stopper.
Everton are looking to now capture him on a permanent basis and, according to Italian outlet Tutto Mercato, are set to kick off talks.
The Goodison Park outfit will speak to Olsen’s agents and Roma, as they look to work out a deal.
Despite Carlo Ancelotti continuing to prefer Jordan Pickford as his number 1, Olsen has been handed chances to impress between the sticks.
He clocked the full 90 minutes against Newcastle United early in his Everton career, while his most recent outing came in an EFL Cup loss against Manchester United.
It remains to be seen what level of fee Roma will be looking for in order to let Olsen depart permanently.
He has a contract in the Italian capital which runs until the summer of 2023, but is surplus to requirements at Roma.