The Toffees' initial plans to move to a new Kirkby home in 2009 were rejected by fans and the respective authorities, who protested the distance from the site to the city centre.
Everton's hierarchy have recently resumed work on finding an alternative venue, with several unspecified locations having been presented to them by the local council. Although Elstone is positive with the early signs regarding the project's viability, he stressed that a decision has yet to be made.
"The stadium remains a big priority," he said at Everton's 2014 general meeting. "We're not ruling out any of the sites that we've identified and that the council has presented to us, but there is one site which is getting more attention and has been getting more attention for a number of months.
"It has been worked on very carefully, diligently and in some detail by not only Everton, but by the council and by advisors, planners, architects, designers, cost consultants, regeneration experts and solicitors. So there's a lot of work going into something that we're very excited about. We need a council being supportive financially and supportive entrepreneurially as well. At the moment there are signs that they are being that, and that's to be welcomed and we're delighted with that.
"We hope it comes to fruition and, if it does, I think it's something that the city and our fans will be very proud of. The board is currently looking not only at this opportunity but also the associated risks and the debt that the club may well have to take on to deliver this, which may or may not be the right thing to do. That's exactly where we are in terms of the project at the moment - that consideration is ongoing by the board.
"But, in summary, there's a degree of optimism - perhaps some cautious optimism - but a degree of optimism. We wouldn't be investing what we're doing without thinking it had a chance of success, and there's a commitment as always to share information with our fans as soon as we can."
Everton have played at Goodison Park, which has a capacity of 40,157, since leaving Anfield in 1892.