The Italy international believes that a double for the Bianconeri is a realistic aim this season, having knocked out reigning European champions Chelsea earlier this month
The Bianconeri won the Scudetto last season and top the table with the winter break imminent, ahead of the likes of Inter, Napoli, Roma, Lazio and Fiorentina.
Last week they beat Shakhtar Donetsk in Ukraine to top a tricky Group E and knock out reigning European champions Chelsea, and Chiellini has revealed that confidence is rising that the double is within reach in 2013.
"We did something special last year winning Scudetto and now we are trying to insist in that way," he told Tuttosport.
"The most beautiful thing is that we are improving game after game: last year someone said we were good just because we didn't player European cups.
"Our first target is Serie A, than we hope to go as far as possible in Champions League. But after recent results, it's logical that we can start thinking about the double. But we must stay concentrated."