Ramos announced last month that he would leave Madrid after failing to reach an agreement to extend his 16-year stay in the Spanish capital.
Magic FM first revealed on Monday that PSG had intensified talks with the centre-back’s camp.
Sources close to the deal said that things have advanced throughout the week and a final decision is expected in the coming days.
Ramos wants a two-year deal but PSG are only offering one with a second year as an option to be triggered by mutual content. His wages should also be lower than what he was earning in Spain and what he would have earned had he stayed at the Bernabeu.