The Blues sacked Graham Potter last weekend after slipping into the bottom half of the Premier League and subsequently made contact with Julian Nagelsmann and Enrique on Tuesday.
Nagelsmann, who has also attracted interest from Tottenham following the departure of Antonio Conte, was sacked by Bayern Munich last month to make way for former Chelsea head coach Thomas Tuchel.
While still a leading candidate, Nagelsmann now seems to have fallen behind the Spaniard in the race to land the Stamford Bridge hot seat.
Enrique has been out of work since his exit from Spain last December, having overseen a disappointing campaign at the World Cup in Qatar.
According to the Independent, Enrique successfully sold a clear vision for the team to Chelsea’s chiefs and is now heading to the capital in a bid to discuss the managerial vacancy in more detail.