However, Clarets boss Brian Laws knows there is a long way to go before the club can be confident the 24-year-old will start the season at Turf Moor.
Eagles turned down a switch to Rangers on Thursday and was immediately linked with Sunderland, with reports suggesting interest from the Premier League side had been a factor in the player rejecting a move to Glasgow.
But Black Cats assistant boss Eric Black told the Lancashire Telegraph: "We have no interest in him."
Eagles has one year remaining on his contract at Burnley and Laws is hopeful of persuading him to stay, saying: "We know his contract is running out and we have to do something about it. There is a market value for a player and I know Chris wants to play at the highest level.
"All those issues were taken into account when we allowed him to talk to Rangers.
"However, he has turned them down and that is great news for us. It gives me another opportunity to talk to him. However, everything will be in his hands and we have to abide by how he feels."