Everton supporters group the Ruleteros Society are trying to arrange a friendly match between David Moyes' side and their Chilean equivalents.
CD Everton are the reigning Chilean champions and celebrate their 100th birthday in July 2009.
They were formed following Everton's visit to Chile in 1909.
Tony Heslop from the group told BBC Radio Merseyside: "It's a win-win thing. Everton against Everton is a great idea."
The Society was formed around five years ago and consists of Everton fans across Britain, with the aim of promoting links between CD Everton and Everton Football Club.
The name Ruleteros (the roulette players) comes from the nickname of CD Everton, who are based in the casino haven of Vina del Mar.
CD Everton are a good middle to lower-table English Premier League side
Tony Heslop from the Ruleteros Society
Heslop added that the Ruleteros Society are exploring as many avenues as possible to make this game happen.
"There might be the possibility of a game between youth teams or a representational 11 in June 2009 out in Chile.
"There could be a strong possibility and we would like to bring it to fruition of a game at Goodison.
"North America could also be an attractive location. It could be a good pre-season warm-up."
Heslop feels that CD Everton would provide stern opposition for the Blues.
"You may have said two or three years ago that they were a good Championship side.
"You would say now with the increased strength of their squad that they're a good middle to lower-table English Premier League side."
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