While sharing a stadium with their Merseyside rivals is a possibility, Kenwright has claimed that the Toffees have been speaking with the Liverpool City Council over building a new ground near Goodison Park.
"We've had a couple of meetings with Liverpool and the Council but it is not moving forward at the moment," Goal.com quotes Kenwright as saying when asked about a ground-share during a General Meeting of club shareholders.
"I agree with you on the obvious advantages it would bring. We have identified a site close to Goodison and we have a council at the moment who are extremely positive in their desire to help us."
Meanwhile, the Reds are said to be planning a redevelopment of Anfield.