You can do this using the Tabulate Intersection (Analysis)—ArcGIS Pro | Documentation tool.
Input Zones Features - your circle buffers
Input Class Features - Your District features
Sum Field - Population number field from your district feature
This would give you the number of estimated persons per intersection of each zone and district.
Each zone may return multiple rows (i.e. 1 for each district it intersects) so you would have to add those row values together.
This is also all based on the assumption that the population is evenly distributed within each district - something you can't avoid unless you have more data e.g. residential dwellings or urban area polygons, then also again you have to make the the assumption that population is evenly distributed within each urban area or residential point and so on and so on...
