Boro enjoyed two successful seasons in Europe from 2004 to 2006, which concluded with them making the Uefa Cup final, which they lost to Sevilla.
"Over the next five years we want to win another cup and we want to be in Europe more often than not," Gibson told a fans' forum.
"We want to make the next 10 years more successful than the last 10. We want to continue making and breaking new records for the club."
Despite spending over £20million themselves this summer, Gibson says Boro will never go beyond their means.
"I'm not concerned with what other clubs do, but right now, we are in 12th place in the Premier League and that tells me we are competing," he said.
"We have spent £23million on new players this summer. When I'm asked about the club, everything seems to be about money, money, money.
"But we have so much else going for us, like the Academy. You don't have to spend millions on players. Players like Julio Arca and Luke Young are proving great value for money."