West Bromwich Albion are also interested in Gibson, a report claims.
The tabloid claims the Toffees privately fear Stones' exit this summer, adding that Manchester City are increasingly confident of landing the England centre-back.
Earlier on Wednesday, the Sun reported that Stones watched his prospective future employers in Tuesday's Champions League semi-final draw against Real Madrid at the Etihad Stadium.
And Everton, who held onto the 21-year-old amid interest from Chelsea last summer, have already begun drawing up contingency plans, according to the Mirror.
Gibson, whose displays have helped Middlesbrough to the brink of promotion from the Championship this season, is described as the Toffees' "number 1 target" and the Merseyside club are said to be in pole position to sign him ahead of West Bromwich Albion.
The 23-year-old has kept clean sheets in almost half of his 31 league appearances and is reportedly valued at around £4 million by the Teesiders, whose chairman, Steve, is Gibson's uncle.
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Left-footed Gibson, who has 10 England Under-21 caps, was previously courted by Everton last summer, the report claims - adding that Boro face a battle to keep him whether they are promoted or not.