With Adriano Basso refusing to budge on signing a new deal at Ashton Gate, manager Gary Johnson is keen to sign goalkeeping cover for Dean Gerken.
The City boss had hoped to sign Begovic on a loan deal but Pompey are not willing to let him leave Fratton Park because their number one keeper David James is currently struggling with a knee injury.
"We were looking at bringing in Asmir Begovic on loan from Portsmouth but they kept holding off going through with the deal," Johnson told City's official website.
"Then on Saturday David James got injured against Arsenal and Begovic came on, so he's no longer an option."