The England left-back has been sidelined for a month after suffering a broken toe, and manager Roberto Martinez expects him to be available to face the Black Cats at Goodison Park.
Bryan Oviedo has proved an impressive replacement in the absence of Baines, and Everton have won four and drawn one of their five matches without the England star.
Martinez said: "Leighton is getting really close. Depending how he goes in training over the next couple of days, he should be ready for Thursday, which is terrific news.
"Of course we have missed him. You can't underestimate the quality of Leighton and his experience, know-how and what that represents.
"We have had an incredible replacement in Bryan Oviedo, he has been terrific. But you want players in good moments of form to be available.
"We are desperate to have Leighton Baines back and desperate for Bryan to be in that moment of form. We need to make sure we use that well for the good of the squad."
Manchester United had an offer for Baines turned down in the summer, and the 29-year-old remains the subject of transfer speculation along with the likes of Ross Barkley and Seamus Coleman.
Martinez is happy for the rumours to start up again when Baines returns to action, but he insisted he will be under no pressure to sell.
He added: "I hope we get more speculation. Now we are going to get Ross Barkley, I am sure, and Seamus Coleman.
"But that is a good sign. The problem is when you are in January and nobody speaks about your players.
"I am quite happy, speculation is welcome. We don't have to sell. We are in a great position financially.
"That is what football should be about - people talking good about your players. It is a good message."