Having been informed via text message by manager Antonio Conte that he has no future at the club, Costa has pushed for a return to Atletico Madrid with reports stating that club are willing to pay in excess of £40m ($51.3m).
But with the La Liga side currently under a transfer embargo meaning that Costa would not be able to play straight away, Goodison Park could be a temporary home for the Spanish international as according to The Mirror, Everton manager Ronald Koeman is open to the idea of a loan switch.
The report states that the Dutchman has been "canvassing opinion behind the scenes" among staff members, gauging whether a player like Costa would be worth the risk.
Despite the arrivals of Sandro Ramirez and Wayne Rooney to compensate for the departure of Romelu Lukaku, the Toffees are said to be looking for a forward to spearhead their attack.
They were previously linked with a move for Arsenal's Olivier Giroud, but the Frenchman decided to stay put at the Emirates Stadium and fight for his place following the signing of Alexandre Lacazette.
And with the transfer window closing in just over a week, a loan deal for Costa would benefit both Everton and Chelsea as the striking options are running out for the Merseysiders while Conte has struggled to offload the Brazil-born forward.
However, the only obstacle is whether the 28-year-old would be receptive to the idea of a move to Liverpool, having previously stated that if he was not to return to Madrid, then he would train in Brazil alone while under a Chelsea contract.