Manchester United have also shown interest in Morata, but the Red Devils look set to lose the battle to the their London rivals as the player would require Champions League football next season.
Morata will certainly move away from Bernabeu in the summer in order to get more playing time before the World Cup in Russia and according to earlier reports Conte had already prepared a sizable bid to bring in the 24-year-old as replacement for the possible departure of Diego Costa.
Madrid are interested in signing Kylian Mbappe, Pierre-Emerick Abaumeyang or Paulo Dubala and should either of those transfers happen, Morata is likely to become a firm bench-warmer.
Morata was sold to Juventus in 2014 but Madrid opted to buy him back two years later. His current contract has a release clause of 80 million, but the price Chelsea could end up paying is likely to be less if the player forces the move. Morata's current contract runs out in 2021.