The Italy international feels the Turin powerhouse must not rest and their laurels and should continue to add to their squad in order to challenge for top honours again next season
The Scudetto holders have been linked with a number of strikers over the past few weeks as they aim to add some more firepower to their squad, but Chiellini feels their attack is not the only area that needs to be strengthened.
"The club is doing its best to improve every area of the team and be in the position to compete at an even higher level," Chiellini told reporters.
"We weren't even in Europe two years ago, yet this season we returned to the Champions League after a lengthy absence and enjoyed a great run.
"We breezed through the last 16 and then came up against Bayern, a team that have objectively shown themselves to be the strongest at this current moment. We now need to continue to improve year on year."
Chiellini is currently preparing for the Confederations Cup with Italy.