The Cherries launched a £10m bid for Begovic but the Blues’ failure to land Craig Gordon from Celtic has seen the move fall part.
Antonio Conte was willing to let Begovic leave in search of first-team football on the condition he signed a replacement.
However, the Daily Mirror report that Gordon has decided to extend his Celtic contract until 2020.
But the Bosnia and Herzegovina international could seal his move away from Stamford Bridge at the end of the season.
Begovic has publicly hinted at an exit and said: “I have spoken with the boss – we are quite open about it, he knows I am desperate to play, and that’s all it is really.
“If he can make it happen for me he will. That is something that is going to be sorted out in the next couple of days.
“I’m 29, 30 in the summer, and feel fantastic and in great shape. It would be a waste not to play and show people what I can do.”