Minister Imam Nahrawi said he had discussed the idea with President Joko Widodo, who is trying to revive Indonesian football after years of underperformance.
"This idea has been discussed... even though it's not easy," Nahrawi said, according to state news agency Antara.
He added the government had also discussed hiring interim Chelsea manager Guus Hiddink.
FIFA last year banned Indonesia from international football, citing government interference in the sport.
The government had refused to withdraw their demands that local football association (PSSI) block two clubs from taking part in the top flight domestic league, because of concerns about their ownership.
The PSSI stood firm but the league was halted after two rounds and eventually cancelled after the government stepped in.
One of the world's most successful coaches, Mourinho said last month that while he had not signed a contract with anybody, he expects to be back in football in time for the start of next season.
"I'm in a position to be open to any club, to any national team, but to be honest if possible I'd like a club, and analyse and think what's best for my career and to be back with everything I have," he said.
Mourinho was sacked by Chelsea in December after the Premier League champions started the season with nine defeats in their first 16 matches.
The 53-year-old, who has also coached Real Madrid, Inter Milan and Porto, has been strongly tipped in media reports to take over from Louis van Gaal at Manchester United next season.
Indonesia, ranked 185th in the FIFA world rankings, are one of four countries vying to host the 2023 Asian Cup.
