The What Works Centre for Children's Social Care, which has been backed with £20m in government funding, has issued the call ahead of its official launch, which is expected to take place in 2020.
The successful bidders will be expected to help test out research programmes aimed at reducing the need for children to come into care and improved supervision and "decision-making" in children's services, the centre's programme director Greg Wilkinson said.
They will also be asked to link up with each other to compile evidence and share it in the sector, with a focus on online media including peer-to-peer networking and toolkits.
"This call is a big and exciting step for the centre," said Wilkinson.
"One of the key lessons from other What Works initiatives is that simply making research findings available is not enough: we need to develop approaches that build decision-makers' and practitioners' capability, motivation and opportunity to make use of evidence.
"We think the best way to develop such approaches is in close partnership with the sector - working with a small group of pioneer localities, using prototyping methods to quickly and iteratively test ideas, and then adapting our approaches in the light of what we learn."
The centre is accepting bids from councils on their own as well as those linking up with other local partners.
Wilkinson added: "The What Works Centre will be successful only if it offers the right balance of support and challenge to do things differently.
"We want partners who will help us find that balance in ways that allow the centre to be characterised by a spirit of working with local authorities - not one of doing things to the sector."
Last month doubts emerged about the long-term future of the centre, which was first announced in January 2016.
Current funding runs out in March 2020 and documents seen by CYP Now revealed that there is "no guarantee of ongoing funding" beyond that.
The centre is being set up by innovation charity Nesta and its research partner is Cardiff University.
A Department for Education spokesperson said: "Special guardians play an important role in giving vulnerable children and young people the care and support they need in a permanent, loving and secure home.
"All councils have a clear legal duty to make arrangements for special guardianship support services such as counselling, advice and financial support where necessary. We have also extended access to our Adoption Support Fund to special guardians of previously looked-after children to help pay for therapeutic services they may need for their child."
