West Ham have increased their offer for Lazio’s Felipe Anderson to 40m euros (£35m) plus add-ons, according to Sky in Italy.
Lazio president Claudio Lotito has asked for 50m euros (£44m) for the midfielder but is likely to accept a 40m euro offer with an easily achievable bonus scheme.
West Ham’s new bid will include a bonus arrangement depending on a number of appearances.
Talks began earlier this month and will continue in the coming days.
One source told Sky Sports News West Ham are hopeful a deal can be done but claims there is interest in the 25-year-old from other clubs.
Since replacing manager David Moyes at the end of the season, Manuel Pellegrini has already signed Issa Diop, Lukasz Fabianski and Ryan Fredericks for the Hammers this summer.
Pellegrini is keen on another midfielder though. It is understood the club have had a £20m offer for Fulham captain Tom Cairney turned down.
West Ham are also set to miss out on Paris Saint-German player Javier Pastore after he arrived at Roma this week for a medical before completing a move to the Italian club.
Sky Sports News understands talks between the Premier League side and Pastore had reopened earlier this month after they had halted due to the player’s high wage demands.