Attracting the Blues to Kirkby would be of enormous benefit to Knowsley council, which is desperately keen to kickstart the town's regeneration.
The proposed site is on land off Valley Road and Cherryfield Drive, partly occupied by Knowsley Community College.
A council spokesman said: "We are in the early stages of discussion with Tesco regarding a potential leisure and retail development close to Kirkby town centre. The development has the potential to incorporate a new stadium for Everton FC.
"Kirkby is one of a number of options being considered by Everton."
Aspokesman for Everton said: "There have been exploratory initial talks, but at this stage no more than that.
"However, it does sound an interesting proposition and we look forward to further talks with Knowsley council and Tesco.
"But we still have several options open to us, and at this stage this is just speculation."
A Tesco spokesman added: "I can confirm we are interested, but it is incredibly early days."
Previous relocation plans have provoked a mixed response from Everton fans, many of whom do not want to leave their historic home.
But recently increasing numbers of supporters have recognised the club may have to leave Goodison to match the financial firepower of their Premiership rivals.
Season-ticket holder Phil Stevens, 24, of Childwall, said: "Kirkby might not be every supporter's first choice, but the club has to be realistic about what is on offer.
"A new stadium is a priority if we are going to compete with the likes of Liverpool."
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