Lanzini was brought in on a season-long loan from the UAE club last July, with an option to make the move permanent.
Sullivan confirmed that the club has signed Lanzini on a permanent deal. His son Jack Sullivan also tweeted that the player would sign a four-year contract.
He said: "Lanzini is a WHU player for the next 4 seasons, with a further 2 year option."
Lanzini has scored six goals in 22 appearances in all competitions for West Ham.
The Buenos Aires-born midfielder made his professional debut for River Plate as a 17-year-old and spent the 2011-12 season with Brazilian club Fluminense before moving to the UAE in 2014.
(Story refiled to fix team to Fluminense in final para)
(Reporting by Aby Jose Koilparambil in Bengaluru; Editing by John O'Brien)
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