More than 160,000 people take part in the series each year. It comprises seven races, including the world’s biggest half marathon, the Great North Run.
The society, which has been the charity partner since 2009, raised £2.2m in the first two years of the deal.
The new partner will receive up to 1,000 places in the series, which is now in its 19th year.
Fiona Vigar, sports marketing consultant at Bupa, said: "This year we are making a change and are looking for a partner that can demonstrate its focus on keeping people well and preventing illness by empowering them to get more active."
An Alzheimer’s Society spokeswoman said that its partnership with Bupa was reviewed every year. "We would have wanted a fourth year, but it would have been unprecedented for Bupa to have a Great Run partner for four years," she said.
A Bupa spokeswoman said: "Every three years, we look for a new charity partner for the Bupa Great Run Series to ensure the benefits of profile-raising and fundraising are shared. Our partnership with the Alzheimer’s Society has been very successful, and we will continue to support it in other projects in the UK, as well as in Australia and New Zealand.
"For example, we will be the lead partner in the Memory Walks the charity is holding in September this year."
Charity applicants are expected to have a track record in securing high-profile celebrities who can promote the partnership and run in the events.
Applications close on 13 May. The successful charity will be announced in September. For more information, go here.