The Icelandic attacking midfielder joins from Hoffenheim for a fee of around £7 million, while Tottenham did not specify the length of his contract.
Sigurdsson spent the second half of last season on loan at Swansea City, who agreed a fee for the 22-year-old in May.
However, the move fell through after manager Brendan Rodgers left for Liverpool.
Villas-Boas was installed as head coach on Tuesday on a three-year deal, replacing Harry Redknapp.
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