The 25-year-old winger scored seven goals and provided 12 assists in an injury-hit season for Lazio last term during which he helped the Rome club to the Europa League quarter-finals.
The move has excited West Ham fans as Anderson has proved to be one of Lazio's key players for a number of seasons, having signed from Santos for £6.75m in 2013.
And according to the likes of The Daily Mail, the East Londoners are in pole position to land Anderson despite interest from a host of top clubs as the Brazilian 'wants to play for Pellegrini'.
But it seems Hammers co-owners David Sullivan and David Gold - who have pledged to finally make good on promises to splash the cash this summer - could be about to leave their supporters frustrated again.
That's because according to the Gazzetta dello Sport West Ham were told Lazio value Anderson at over £40 million but lodged a bid £10 million short of that figure which was promptly rejected.
While it is completely acceptable that West Ham's owners would try to get the player for the best possible price, their track record shows they have a habit of upsetting other clubs by submitting derisory bids for players.
Last summer Sullivan and Gold became embroiled in a very public spat with the owners of Sporting Lisbon who claimed West Ham were never serious about signing William Carvalho.
Sullivan released an email claiming to show his bid for the player which fell over £10 million short of Sporting's much-publicised asking price.
It all got very messy and almost resulted in legal action with Sullivan and Gold called some unsavoury names along the way, as reported by the likes of the Evening Standard at the time.
Hammers fans won't have a problem with the club haggling but not in the public domain and at some stage West Ham are going to have to stump up the cash if they truly want to sign top players.