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The 23-year-old, who plays for Swedish side Halmstads, has already attracted admiring glances from Premiership duo Birmingham City and Everton but it is believed that Millwall are the only club to have made a firm offer for the player.
Thorvaldsson has been capped five times by Iceland and finished last season as Sweden's top goalscorer with 16 goals in 16 goals games.
Should all go to plan, he is expected to sign for The Lions by the end of the week.