The Dutchman wants Napoli striker Manolo Gabbiadini as well as Manchester United duo Morgan Schneiderlin and Memphis Depay after telling the Toffees’ major-shareholder Farhad Moshiri that his squad needs an overhaul.
Everton have slipped into midtable following a promising start to Koeman’s reign and he has warned that he will need at least three new recruits in the January transfer window.
A summer bid for Gabbiadini, 25, was rejected by Serie A side Napoli, but the Blues have been given fresh encouragement that he is now available for £18m.
Koeman wants Gabbiadini to take some of the strain off Romelu Lukaku, who has scored nine of the Toffees’
21 Premier League goals this season.
Everton are also confident of doing business with United for midfield enforcer Schneiderlin and winger Depay.
Both players have been told they can leave Old Trafford by Jose Mourinho.
And while United will want to recoup most of the £24m they paid Southampton for Schneiderlin 18 months ago, Depay is available for just £15m after a disastrous time in Manchester.
Koeman is desperate to add both power and pace to his squad and needs Depay after losing Yannick Bolasie for the season with a knee injury.
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Everton have already been boosted by the news that Lukaku is ready to sign a £100,000-a-week contract that will make him the club’s highest-paid player.
But the Goodison powers know that they will have to bring in some fresh blood to pacify fans who fear another season of under achievement.
Roberto Martinez was sacked last summer despite taking Everton to the semi-finals of both the FA Cup and League Cup.
And while Moshiri promised Koeman a significant transfer budget when he arrived to take charge from Southampton, the 53-year-old has found it tough to land players with the right quality to improve his squad.
Koeman is also ready to listen to offers for Tom Cleverley, Darron Gibson, Arouna Kone, Bryan Oviedo and Oumar Niasse in a bid to clear the decks for new arrivals.