The Serbian is out of favour with Sporting coach Paulo Bento, after joining the Portuguese side in 2007.
The keeper's agent Zoran Stojadinovic says if all goes well, Stojkovic will be offered a year's loan deal by Everton boss David Moyes with the option of making the move permanent.
"If the coach is impressed by Stojkovic, he may decide to move towards purchasing the player," Stojadinovic said.