The 23-year-old, who joined West Ham in July, has netted five goals in the league and will complete the deal when the transfer window reopens.
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 Fluidness before moving to the UAE in 2014.
