The Scottish champions recently completed the signing of goalkeeper Vasilis Barkas from Greek giants AEK Athens.
However, they have made an enquiry for Olsen, according to Sky Italia, looking to take him to Celtic Park on loan.
It is claimed that the Scottish champions asked to take Olsen on a loan deal, without paying a fee, while his salary would also have been shared between the two clubs.
Roma considered the conditions attached by Celtic to the loan to be unacceptable and rebuffed the proposal.
Olsen, 32, spent last seaaon on loan at Serie A side Cagliari, where he played regular first team football.
The Giallorossi splashed in the region of €12m to sign Olsen from FC Copenhagen in the summer of 2018 as they looked to replace Alisson, after selling him to Liverpool.
The Sweden international put pen to paper to a five-year contract which he joined Roma.
