The Italian was sacked in the summer after two years in charge at Stamford Bridge, with Maurizio Sarri installed as his successor.
A fifth-placed finish and failure to qualify for the Champions League brought an end to Conte's reign, but he won the Premier League in his first season in charge, and also lifted the FA Cup last term.
Madrid initiated contact with Conte in the summer, before their headline-grabbing appointment of then-Spain coach Lopetegui prior to the start of the World Cup.
However, Conte has yet to receive any compensation from Chelsea for his departure from the Premier League club, which came after the start of pre-season training.
The Italian was earning £9 million ($12m) per season in west London, and is prepared to take legal action if he is not paid the full amount in compensation.
Conte is, therefore, wary of accepting any job offers prior to him receiving a pay-out, as it may impact any settlement.
Madrid are aware that they will need to make an attractive offer to Conte to ensure that they can appoint him if they decide to dispense with the services of Lopetegui.
The Champions League holders are enduring a tumultuous period after going four games without a win and a goal.
"We as players are with him to the death," the defender told EFE.
"We are calm. We have been through situations like this and worse, and we've always been able to get out of them.
"I can't guarantee we will win every title but Real Madrid have always proven that with the coach and players that we have, we will always be fighting for everything.
"We will be there until the end."