Juventus sporting director Fabio Paratici is currently looking at thinning out his squad, having spent over €100 million (£90m/$111m) on new signings including defenders Matthijs de Ligt, Luca Pellegrini and Merih Demiral.
They have recouped €29.5m (£27m/$33m) from the sale of Leonardo Spinazzola to Roma, but now wish to offload attacking talent.
Gonzalo Higuain, Mario Mandzukic and Kean are all on the chopping block, with 19-year-old Kean having emerged as a target for the Toffees.
Sources close to the deal have told Goal that the Goodison Park outfit have already been in contact with the Bianconeri regarding the potential transfer of the forward and are not deterred by the asking price of €40m (£36m/$44m) including bonuses.
Despite both clubs being happy with the transfer fee, negotiations have slowed down as Juventus were insistent on placing a buyback clause in the agreement - something which Everton are vehemently opposed to.
Kean's agent, Mino Raiola, is also involved in the talks, but Everton are hopeful that a deal can be reached between all parties and will resume talks on Monday.
Everton are looking strengthen their attack this summer, having sold winger Ademola Lookman to RB Leipzig for £22 million ($27m).
Kean is seen as one of the top talents in Italian football, having scored six goals in 13 Serie A appearances for Juventus last season, while also breaking into the Italy national team with two goals in his first three appearances.
Toffees boss Marco Silva said he was eager to get in new players before the start of the Premier League season, telling reporters last week that he wanted to get their transfer business done quickly.
“It should be if you really want to be competitive and to do better than last season," he said. “Because at the end of the day, it is what we want to do: do better than last season."
Silva also looks likely to have to replace midfielder Idrissa Gana Gueye, who has been linked with a move to Paris Saint-Germain, but is waiting to see if that move materialises before bringing in a new midfielder.
“Of course, we don't know what will happen with Gana and there is a lot of speculation but let's see what will happen in the next few days," he continued.
“If Gana goes then we have to replace him directly and one player has to come in with that profile and, I said to you, one clear No. 6 to come and to play."
Mainz midfielder Jean-Philippe Gbamin has been touted as one possible replacement for Gueye, but Everton's immediate concern is the arrival of Kean, with talks set to continue today.