Olsen made it clear to Swedish media during the national team gathering in Marbella in March that a move home was possible, but that nothing was clear.
His contract with Aston Villa expires this summer and Olsen has been free to discuss with other clubs since the turn of the year.
According to Aftonbladet, Olsen and Malmö FF have agreed on a longer contract and he will join the club this summer, when the Allsvenskan transfer window opens. It is unclear when he is expected to be officially presented.
Olsen left Malmö FF in July 2015 after winning two Swedish Championship titles and one cup title. He has since played for PAOK Saloniki, FC Copenhagen, Cagliari, Sheffield United, Everton, Roma, and Aston Villa.
Olsen has 76 caps for Sweden. The latest was against Northern Ireland at Strawberry arena at the end of March.
