The Hammers are keen to get their business underway before this summer's World Cup, as new manager Manuel Pellegrini looks to instigate a complete overhaul of the playing squad at the London Stadium.
MailOnline Sport have now revealed that the Hammers have made a bid for the Brazilian, days after reports first linked the winger with a move to the Premier League.
Anderson is understood be keen on working under Pellegrini, and wants to get the move over the line as soon as possible.
The 25 year-old has been one of Serie A's standout performers in recent years, but injuries hindered his progress last season, with Anderson making only 21 appearances in the league. He did score four goals and contribute eight assists in that time though, showing the quality that has seen him linked with a move abroad before.
His style of play is likely to suit Manuel Pellegrini, with the Chilean known to prefer an attacking, fast paced style of play. With that in mind, Anderson could become the first piece of the jigsaw as Pellegrini looks to assemble a side capable of pushing for Europa League qualification.
The Hammers are also reportedly close to signing Barcelona defender Marlon Santos, and highly rated Fulham right back Ryan Fredericks has reportedly agreed a four-year deal at the London Stadium.