Sevilla's Jules Kounde is understood to be at the top of Chelsea's centre-back wishlist, but Thomas Tuchel is thought to want more than one defensive addition and Bremer is another of the players on his radar.
Transfer expert Gianluca di Marzio claims that Chelsea have lodged an 'important proposal' for the 25-year-old in an attempt to beat their rivals to his signature.
However, Inter are reluctant to meet Torino's £42.6m asking price, and are also unwilling to include Cesare Casadei as part of the deal.
Liverpool, Paris Saint-Germain, Leicester City and Newcastle United are among the other clubs to have been linked with Bremer in the past, with the defender drawing attention with a string of impressive performances last season.
The report claims that Chelsea have now stolen a march in the race for his signature, though, while Bremer himself has confirmed that he plans to leave Torino.
The Brazilian told ESPN Brasil: "I told the coach and the management [last summer] 'I want to change now'. So they told me: 'Stay this year, help us. Next year you can take the next step'.
"The fans already know that I have this desire to play in the Champions League, to go there to earn a call-up to the Brazilian national team.
"I need to play at a much better level, so I think it's a matter of time. I'm analysing some proposals, but I think that next season I will no longer play for Torino."
Bremer made 33 appearances in Serie A last season, scoring three goals to help Torino to a mid-table finish.
Chelsea, meanwhile, are expected to undergo a significant summer overhaul on and off the pitch after being taken over by a consortium led by Todd Boehly.