There are two meanings to the word circle. Among non-mathematicians, it often means the curve together with its interior. Among mathematicians, circle refers to the curve, and disk refers to the curve together with its interior.
Using the standard axiomatization of set theory (ZFC), it can be proved that no paradoxical decomposition of the circle exists, nor of the disk.
However, there is a famous related result, called the Banach-Tarski paradox, that says that the $3$-dimensional ball of radius $1$ can be decomposed into a finite number of sets, which can then be reassembled to make two complete $3$-dimensional balls of radius $1$. There is a large family of related results. The one that is closest to your question is that for any $r$ and $R$, a ball of radius $r$ can be decomposed into a finite number of sets which can be reassembled to make a ball of radius $R$.
Unrelated to your question, but interesting, is the following beautiful theorem of Laczkovich.
The disk of area $1$ can be decomposed into a finite number of sets which can be reassembled to make a square of area $1$.