The 21-year-old is yet to start for the Foxes in the Premier League as he came on from the bench in the first three fixtures. He had only nine starts to his name, while making 21 appearances from the bench in the 2016/17 season.
According to the Mirror, Gray is frustrated with the lack of playing time at the King Power Stadium and is looking for a move away from his current employers. His representatives Luca Bascherini and Kenny Brady were at the Foxes' training ground on 29 August, trying to resolve his future.
The Eagles were ready to offer Andros Townsend as part of the deal. However, the deal is no longer believed to be on the cards. Despite the offer from Bournemouth, the north London club are ahead of the other two Premier League clubs in the race to sign Gray.
Leicester registered a comfortable 4-1 victory over Sheffield United in the Carabao Cup. Gray scored one and registered an assist as the Foxes progressed to the third round of the cup competition.
Apart from Gray, Islam Slimani and Ahmed Musa are also linked with a move away from the club. Leicester manager Craig Shakespeare said that he expects to retain the trio, despite the three players attracting interest from other clubs.
"It was important to give the fringe players, or those who haven't played, some game time and when you do that you hope they'll respond in the right way," Shakespeare told Sky Sports.
"I thought the team responded really well. They're part of the squad and they carry on. We'll need everyone as we go along because it's a tough old season in the Premier League and whether it's a loss of form, a change of tactics or injuries, you have all that to contend with."
Gray could continue at the club if Riyad Mahrez finds a new club on the final day of the summer transfer window. Algeria have allowed Mahrez to skip national team training to complete a transfer.