Celtic are desperate to keep Tierney but a host of Premier League clubs are seeking his signature.
And the Telegraph claim Everton are planning a mega bid for the full-back.
Tierney, 21, has cemented his place in the Celtic first XI after breaking into the team from their youth ranks four years ago.
The Scotland international has also been linked with Manchester United in recent weeks.
But Everton are readying an offer that would smash Celtic’s record sale of a player.
The Hoops sold Victor Wanyama to Southampton for £12.5m back in July 2013.
But Tierney could fetch double that.
The Telegraph claim Toffees manager Marco Silva wants a long-term replacement for Leighton Baines.
And Tierney has been earmarked for his breakthrough season last campaign, where he picked up the Hoops captaincy on a number of occasions.
Brendan Rodgers wants Tierney to stay in order for Celtic to be more competitive in Europe next season, while a title bid from Aberdeen and Rangers could be on the cards.
Asked about his future back in April, Tierney said: “I’m not thinking about anyone other than Celtic just now.
“I’m enjoying my career here.
“I’ve just signed a six-year deal, so if that doesn’t say long-term, I don’t know what does.”