Chelsea chief Abramovich has left Conte in the lurch ever since the Blues’ FA Cup win over Manchester United last month.
Conte has been widely tipped to leave his post at Stamford Bridge having failed to steer them to the Premier League title last term.
Enrique has emerged as a potential successor, with the Spaniard still out of work having quit Barcelona a year ago.
And Spanish outlet Don Balon say Chelsea owner Abramovich has confided in Enrique about his Conte plans.
The Italian arguably deserves another season to make his mark.
He won the club the league title in 2017, tidying up the mess left to him by current Manchester United manager Jose Mourinho.
And Don Balon say Abramovich is now sensationally ready to let Conte stay on at Chelsea.
He supposedly thinks that, by sacking the former Juventus boss, he would see himself out of pocket.
As a result, he is now reportedly ready to forget Enrique and stick with his current boss.
Meanwhile, speaking last month, Conte insisted he was ready to stay at Chelsea for another 12 months.
“After these two seasons I will always love this colour, these fans, this club,” said Conte.
“Also if my future must be in a different way.
“I have a contract and I’m committed to this club, despite a lot of speculation.
“One day I had the possibility to go to the national team, one day this team, one day another important team. In every press conference I always say I’m committed to this club, to respect my contract.
“You can tell what you want, but I’m a serial winner. I showed this in England in a difficult moment for the club, after 10th place.
“I’ve showed this also today, because we found the right way to win this trophy. I’m very happy for the players, I’m very happy for our fans.
“At the same time I’m happy for the club. The decision to charge me, in two seasons I brought two trophies for this club.”