Sigurdsson helped Iceland to four games unbeaten ahead of their quarter-final defeat to France
Tottenham Hotspur, Leicester City and Liverpool were all linked with moves for the defender last week, following a string of impressive displays at Euro 2016.
Sigurdsson was on target against England
According to the Sun, Sigurdsson, who plays in Russia for FC Krasnodar, could be available for as little as £4 million, while his reported £11,000-a-week wages make him a potential bargain for someone.
Everton, who were also linked with Netherlands international Bruno Martins Indi on Tuesday (O Jogo), could be in the market for a new centre-half, with John Stones said to fancy a move to Manchester City (Daily Mail), Ramiro Funes Mori off to the Olympics and Phil Jagielka due to turn 34 next month.
Stones could be bound for Manchester Cty
Middlesbrough and Watford, on the other hand, have both added to their defensive ranks in recent weeks, landing Bernardo Espinosa and Christian Kabasele respectively.
However, Tottenham, who recently lost Jan Vertonghen to injury, Leicester, who are working on a protracted deal to sign Burnley's Michael Keane (Lancashire Telegraph), and Liverpool, who have already released Kolo Toure with Martin Skrtel set to follow, could all pose more serious threats to Everton's hopes.
Vertonghen's injury could prompt Spurs to enter the market