The Blues are reported to have signed the Spanish international for a £21 million fee according to the BBC.
Manchester United were closely linked with Pedro as a replacement for Angel Di Maria who joined PSG for £44.3 million.
Pedro scored the winning goal in the 5-4 victory over Sevilla in the UEFA Super Cup final and has a total of 99 goals in 326 appearances for La Liga champions.
The arrival of Turan is believed to have had an impact on Barcelona allowing Pedro to leave. However, the bearded winger will not be able to play a professional game for the Champions League title holders until January due to the transfer ban.
Chelsea were also closely linked with Turan prior to his move to Barcelona following his agent Ahmet Bulut claiming his client would be joining a Premier League side playing Champions League football.
“He intends to leave Spain this summer. It is his will to go out to play in the Premiership, that is where he would like to continue his career at this time,” Bulut told AS.