The Bees hope to leave their 106-year home for a temporary home in Edgware at the start of next season.
And Kleanthous is adamant the club need a modern stadium to help enhance the club's reputation, which has already been improved due the arrival of Edgar Davids.
"We need to have facilities that are in the 21st century," he told the Barnet Times. "Underhill is an old stadium and it needs a lot of work. We are constantly patching up to keep it safe. But we need to modernise.
"The lease (which allows access to the south side of Underhill) expired at the end of December and there is nothing more we can do on that.
"The council have given planning permission to us to build a new entrance to the stadium on that side but you can't access it without the lease.
"So nobody is going to spend several more millions of pounds building a new stadium that you can't then access. So we have no choice but to move on.
"What we are building (at The Hive) is a small permanent stadium but the move is intended to be temporary because it will be too small for our use going forward.
"This season we turned a corner. We went from being the club that nobody wanted to play for, to the club that everybody wants to play for.
"All of a sudden the way the football world looks at the club has changed. That is because we have got away from the old crumbling image of a club in an old facility in a borough that didn't want us to advance."