Ronald Koeman confirmed this week that he would be open to taking the stroppy Brazilian to Goodison Park.
Mirror Sport understands that Koeman has been canvassing opinion behind the scenes among his staff as to whether it is worth taking a punt on the striker.
Costa is currently in a stand-off with the Premier League champions and there little chance of boss Antonio Conte accepting him back into the fold.
He wants to join Atletico Madrid – a switch that cannot take place until January because of the transfer embargo in place at the Spanish club.
But Koeman wants to take him in now and is willing to sanction a temporary deal.
Everton’s boss is still seeking a spearhead for his attack in the wake of Romelu Lukaku’s £75m defection to Manchester United.
He had high hopes that he could persuade Olivier Giroud to leave Arsenal but the France striker appears unfazed by the arrival of Alexander Lacazette and is ready to fight for his place.
However, it remains to be seen whether the troublesome but gifted old-school centre-forward would consider a switch to Merseyside.