Chelsea has made moves to sign the 26-year-old but is wary of paying over the odds while they are also running parallel negotiations for striker Alvaro Morata who is their priority.
The Spain global must now agree on personal terms and pass a medical to complete the transfer.
Morata, 24, had been heavily linked with Manchester United but the Red Devils could not reach a deal with Real.
Chelsea were in the market for a striker as Spain's Diego Costa is expected to leave the club having not travelled on their pre-season tour to China and Singapore.
The fee is thought to be in the region of £70 million.
Over 500000 passports issued so far this year - Coveney
Online renewals now account for nearly 17% of all applications, well ahead of initial projections. People from Britain must prove a close family link to an Irish citizen.
The deal comes a day after Blues manager Conte signed a new deal at the club.
Morata has previously spoken of being "indebted" to Conte, who brought the youngster to Juventus, but never coached him due to taking the Italy job.
So, in the end, Conte got his man.
As per the Daily Mail, Chelsea would rather secure the services of Morata than Borussia Dortmund speedster Pierre-Emerick Aubameyang or Torino goal-machine Andrea Belotti. However, questions remain. £70 million of thereabouts represents quite the risk for a man who has no Premier League experience and was not a guaranteed starter for Real Madrid a year ago.
Speaking on Wednesday, Mourinho said he had little interest in his former club's movements as he challenged Lukaku to prove his worth at the top level, including the Champions League.
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