Local devolution is considered a policy area where there is scope for some common purpose between the Scottish and UK governments.
Both have said they are willing to work together on pushing power into city regions and local communities.
But Swinney has warned the prime minister not to overstep by stripping power from Holyrood.
The first minister is expected to promote his renewed push for independence in his keynote speech at the SNP conference on Saturday.
Swinney previously welcomed Burnham telling MPs that a border poll would be considered in both Scotland and Northern Ireland if there was a clear consensus and public opinion had changed.
But Burnham later pushed back on that by telling Swinney at a meeting in Glasgow that a second Scottish independence referendum remains “off limits”.
Meanwhile, Conservative leader Kemi Badenoch has said the Tories currently have “no plans” to take away powers from the Scottish Parliament – but that “maybe we need a review” into how devolution is working.
Speaking during a visit to Edinburgh on Thursday, she said: “Devolution is a process. It’s not a religion which we should worship.
“This is a process. If a process isn’t working, how can you improve the process rather than just say the process is there, there’s nothing we can do? We are about solutions.”
Badenoch added: “We don’t have any plans to take powers away. What we do have plans for are how to make people’s lives richer.”
