Sure it's simple. You need to use @@ or center to get the center point and then @> to use contains.

SELECT
  x AS r_value,
  circle('0,0', x) @> @@ circle('2,3', 1) AS contains
FROM generate_series(1,5) AS gs(x);

-or-

SELECT
  x AS r_value,
  circle('0,0', x) @> center(circle('2,3', 1)) AS contains
FROM generate_series(1,5) AS gs(x);

This brute-force tests circles with the center at the origin, with a radius of [1,5] whether they contain the center of circle('2,3',1).

See the docs on geometry functions for more info.

select * from circles a , circles b
 where st_intersects(a.geom,st_centroid(b.geom)) 

Also, you can filter the results by adding condition in where clause. I am assuming main circle name.

select * from circles a , circles b
 where st_intersects(a.geom,st_centroid(b.geom))  and a.circle_name = 'Main Circle';