Even though Tottenham have spent almost £60million on new signings this summer, Villas-Boas is still looking for new additions to his squad as he looks to secure Champions League qualification.
The Tottenham manager was interested in signing Willian when he was at Shakhtar Donetsk last season and the Portuguese was also linked with the Brazilian when he was in charge of Chelsea.
Anzhi owner Suleiman Kerimov is looking to sell most of the club's highly paid stars in order to shrink their vast wage budget.
At a press conference to announce Tottenham's support for NHS campaign 'Get to know cancer', Villas-Boas was asked whether he had made a bid for the 25-year-old former Shakhtar player. He replied: "No, no."
When asked whether he admired the player, the Tottenham manager said: "Yes. This is a player who I have spoken about in the past, not very long ago. He is a player who I appreciate. He belongs to them and I respect them so I wouldn't like to extend myself too much."
Liverpool are also thought to be keen on Willian's signature.
Tottenham broke their transfer record twice to sign Paulinho and then Roberto Soldado, while Nacer Chadli and Etienne Capoue have also joined in the last few weeks.
Yet it was clear from Sunday's 1-0 win at Crystal Palace, which came courtesy of a Soldado penalty, that the team are far from the finished article.
Spurs are light in defence following the sale of Steven Caulker and Villas-Boas concedes there is a good chance of at least one more player arriving before the transfer window shuts on September 2.
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- Shakhtar Donetsk