The public broadcaster admitted it anticipates the decision will "lead to the loss of some existing rights and events".
BBC's sports live coverage had already been scaled back this year after it lost the rights to The Open golf championship to Sky Sports, announced it would be sharing Six Nations coverage with ITV and - from 2022 - rights to the Olympics which were acquired by Discovery, owners of Eurosport, for £920m earlier this year.
A BBC statement said:
We have already made some tough choices which have contributed to the savings, for instance around the Open golf. However, we have also recently secured a series of important rights - including Wimbledon, Premier League highlights, live coverage of Euro 2016 and 2020 football championships and Six Nations rugby shared with ITV.
Yet the BBC now looks likely to lose further live rights amongst increased competition from rivals such as Sky, BT, Discovery and Qatari-owned beIN Sports all wielding significant financial clout.
Deals to show live Formula One and FA Cup football are both set to expire in 2018.
The £35m cut to the BBC's sports budget forms part of a wider cost-cutting regimen as the corporation looks to make up a £150m funding shortfall due to a "loophole" in how the licence fee is collected and reach the government's target of £550m savings by 2020.