The Goodison club have already rejected a £57m bid for the Belgian striker, who has told friends he wants to return to Stamford Bridge this summer.
But Conte is confident Chelsea’s second bid will tip the scales in their favour – even though the Toffees are determined to get close to their asking price of £75m as possible, as they have insisted with Manchester City target John Stones.
Lukaku, still only 23, feels he has unfinished business in west London where he played alongside countrymen Eden Hazard, Thibaut Courtois and Michy Batshuayi.
Conte is looking for the ideal replacement for current Chelsea frontman Diego Costa, who will this week be the subject of a £40m offer from his former club Atletico Madrid.
The Italian manager is understood to be hoping to hang onto Costa despite the Spain international’s difficult season last year.
And despite having already spent £33m on Michy Batshuayi, Conte wants to add the depth that Jose Mourinho so badly lacked at Stamford Bridge up front, with none of Loic Remy, Radamel Falcao or Bertrand Traore providing adequte back up to the misfiring Costa.
Ronald Koeman meanwhile will be equally desperate to keep Lukaku at Goodison.
The Belgium striker has scored 43 league goals in his three seasons at Everton and will prove difficult to replace, especially with so much of the transfer window already having passed.
And if the Dutch manager is to lose Stones to City, then it is expected only to strenghten his resolve to retain the services of Lukaku.