At the most recent international window in March, the veteran hinted that a move back to Sweden was a possibility, but nothing was set in stone.
The same sources claimed that in recent times, a return to either Malmo or another of his former clubs, FC Copenhagen, were rumoured to be in the pipeline, with the Danish side having a substantial offer turned down last summer.
But the call from home appears to have been too strong, and the 35-year-old will return to the club where he began his youth career, then played professionally from 2012 to 2015.
Sportbladet suggest that a deal has been agreed, but nothing can be made official until the transfer window opens in July.
Olsen has not played in the Allsvenskan since departing MFF in 2015, and has since played for PAOK, Copenhagen, Roma, Cagliari, Everton, Sheffield United and - since 2022 - Aston Villa, for whom he has been back-up to Emiliano Martinez and made 21 appearances in all competitions.
Four of those appearances have come this season, three in the Premier League, but none in Villa's run to the UEFA Champions League Quarter-Finals.
Olsen won his 76th cap for the national team last month, overtaking the number of appearances he has made for Malmo, with whom he was twice an Allsvenskan champion.
The Himmelsblatt have made an underwhelming start to their title defence so far in 2025, losing 2-0 to Hammarby on Wednesday to extend their winless run to three matches.
Brazilian goalkeeper Ricardo Friedrich is currently deputising for the injured Johan Dahlin, who was Olsen's teammate during his previous spell at the club.