Elementary Darboux transformations of Bessel functions are discussed. In Theorem 1 we present an improved version of a general factorization approach which goes back to E. Schrödinger, in terms of the two interrelated linear differential substitutions $B_1$ and $B_2$. The main Theorem 2 deals with the Bessel–Riccati equations. The elementary Darboux transformations are reduced to fraction-rational ones. It is shown that a fixed point of the latter generates the rational in $x$ solutions of Bessel–Riccati equations introduced by Theorem 2. It should be noted that Bessel functions are considered as eigenfunctions $A\psi=\lambda\psi$ of the Euler operators $A=e^{2t}\left(D_t^2+a_1D_t+a_2\right)$ with constant coefficients $a_1$ and $a_2$. This enables one (Lemma 3) to build up asymptotic solutions of the Bessel–Riccati equations in the form of series in inverse powers of the parameter $z=kx$, $k^2=\lambda$, $x=e^{-t}$. It is also shown that these formal series in inverse powers of the spectral parameter $k=\sqrt \lambda$ are convergent if the rational solutions of the corresponding Bessel–Riccati equation from Theorem 2 are exist.