With doubts over the long-term future of Ross Barkley - who is still yet to sign a new contract and could be sold this summer as a result - Ronald Koeman is keen to bring Sigurdsson to Goodison Park, but he may have to rethink his options following the star's latest comments.
Koeman issued an ultimatum to Barkley earlier in May, urging him to sign a new contract or make clear his decision to leave.
Asked on Friday if he was confident the England global would stay, Koeman said: "No". Okay, I'm not anxious because I like to work with players who like to stay.
"We will see what the answer will be - and I don't know what will happen if he doesn't give any answer about that".
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Everton's season concludes with Sunday's Premier League trip to London to face Arsenal and asked on Friday whether there was any update on Barkley's future, Koeman said: "I don't know".
"We will go on and we're looking for players in that position".
'If you offer a player a new contract, and a good contract, that means you want to keep the player.
'If we can do our business the team for next season will be really strong.
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Koeman says the length of time it has taken Barkley to consider the deal suggests he could move on.
Harry Kane's four-goal haul at Leicester City on Thursday sent the Tottenham striker two goals ahead of Everton's Romelu Lukaku in the race for the Premier League's Golden Boot award. "It will be hard but we will try to help", added Koeman.
"Even if he doesn't win the Golden Boot, 24 is a good number of goals".
'I have got a picture of where we need to improve.
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