It is also the largest fee that Chelsea have ever paid for a midfielder, beating the £32m paid to acquire fellow France worldwide N'Golo Kante from Leicester City last July.
According to reports, Chelsea and West Ham are involved in a battle to sign Javier Hernandez.
New Chelsea signing Tiemoue Bakayoko believes he has joined the "biggest club" in England.
The English champion didn't disclose the fee for the 22-year-old central midfielder, who has signed a five-year deal. "Signing was the natural thing for me because it is a club I loved very much in my childhood", Bakayoko said.
Last season, Costa was clearly the favoured striker, with Chelsea's second choice forward, Michy Batshuayi, only starting one game all season, whilst making 19 appearances from the bench, demonstrating that Conte will be looking to buy big in order to replace their star striker.
Bakayoko, formerly of Monaco, was a mainstay of the french club's midfield throughout their Ligue 1 and Champions League campaigns last season. In his time at the Bridge he has scored 52 goals in just 89 Premier League games, helping them to two Premier League titles. "When he arrived at Monaco I wasn't a very, very good player, but I had discussions with him, he gave me advice, and he allowed me to play my style of football more simply".
"I am very happy to be here and to join this great team", Bakayoko told the club's website.