The Brazilian midfielder has emerged as the top target for David Moyes as he seeks to add more creativity to his squad in the coming days.
His representatives are in London, holding talks with West Ham but negotiations with Lyon have been trickier.
The French giants were quick to reject West Ham’s opening bid of €50m and are holding out for more money.
According to French sports daily L’Equipe, the Hammers have gone straight in with another bid for Paqueta on Thursday.
The east London club sent a new offer worth €42m and another €15m in potential add-ons as they push to get their hands on the player.
Lyon are yet to respond to the improved bid from the Hammers but the positions of the two clubs are getting closer.
The new offer is close to the €60m Lyon want from the potential departure of Paqueta in the final week of the window.
West Ham are still to discuss personal terms with the player but for the moment they are focusing on reaching an agreement with Lyon.