Harris, who first became involved in the club back in 1997 as a shirt sponsor, has overseen the club's rise to become the most successful club in Welsh domestic football.
For years the club has competed in Europe - and this season returned to the Champions League after a two year absence.
However, they suffered late heartbreak last week, losing 2-1 on aggregate to Northern Irish champions Linfield - and they will now play Vikingur Reykjavick of Iceland in the UEFA Europa Conference League.
Last year the club were within touching distance of the Conference League group stages - and Harris believes they can reach that level in the next three years.
Speaking to BBC Radio Wales for a special show on 30 years of the Cymru Premier - the Saints owner also said he is hoping to find a new site in Oswestry to build a 10,000 seater ground.
The idea has previously been floated by the businessman but no plans came to fruition.
Harris explained: "Getting to the group stage is very, very difficult in the Champions League.
"But I believe getting into the group stages of one of the three competitions is well within our grasp in the next two or three seasons.
"Our ability to compete off the pitch is growing but if you also look at what we're doing within our academy, we're attracting the best young kids in the local area.
"We're hoping to find a new site in Oswestry to build a new ground with hopefully 10,000 seats.
"My business in America is going very, very well and I intend to invest heavily in the New Saints to achieve those goals."
Also on the show, Harris explained how he wants to see the Cymru Premier become a fully professional 12 team league.
It has been reported that the FAW is looking at an idea of turning the league professional, with ten teams playing summer football.
Harris added: "I'd love to see the league full time," Harris told BBC Sport Wales.
"Change for nothing is not worth changing and I'm a big believer that any change from here has to be backed by financial plans from the FAW.
"Without full time there's no chance I would ever support any change to the league.
"Twelve seems to be a good league, not as good as it would be if it was 16 or 18, but at least there's enough money to bring good football."