Chelsea are aware that Hazard would be interested in a move to Real Madrid, but the Spaniards are yet to make a move for the Belgian as they explore a deal for Paris Saint-Germain star Neymar.
Despite the fact the club look set to miss out on Champions League qualification, there remains a hope at Chelsea that Hazard could sign a new contract if Real focus their attention elsewhere.
And Abramovich is aware that a move for a more attack-minded coach, such as Sarri or Enrique, would boost his bid to convince Hazard that his long-term future remains at Stamford Bridge.
Having initially targeted former Barcelona manager Enrique, Chelsea turned their attention to Sarri, who has guided Napoli to second place in Serie A.
But sources close to the process insist both men are still in the frame with the Chelsea board yet to make a final decision on either man or find a compromise with Conte over his pay-off.
Out-of-work Enrique wants big wages and a sizeable budget, while Sarri would cost £7m in compensation to Napoli and there are concerns over his temperament.
(Credit : Telegraph uk)