The 40-year-old is his country's first-choice goalkeeper but looks set to be behind Petr Cech in the Blues' pecking order after moving to Stamford Bridge on a free transfer.
But Osieck told reporters: "What [Schwarzer] told me was, when he agreed to sign for Chelsea, his signature was based on conversations with Mourinho, that he obviously promised to give him sufficient playing time because of the great number of games Chelsea has to play, in the league, in various cups and of course in Europe.
"Mark was very confident that there was a lot of honesty in Mourinho's words."
|200/1||Australia are 200/1 with Bet365 to win the World Cup
They are among a host of players pushing for Schwarzer's spot, including Liverpool man Brad Jones, Reading's Adam Federici, Mitch Langerak of Borussia Dortmund and Mat Ryan, who switched from Central Coast to Belgian side Club Brugge at the end of last season.
"I have three goalkeepers here and a couple more [to chose from]," Osieck added. "Langerak, with all his talent, he needs to play. He's definitely a good keeper.
"Mat Ryan obviously made the first team of Brugge. Then we have a couple more in England, there's Federici and Jones and then we have the domestic boys here. Mark Birighitti has shown promise and Nathan Coe had a good game against Liverpool [for Melbourne Victory].
"Eugene has already showed what he can do - in particular the first game, when he was outstanding - so it's going to be a tight race."