This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in $\mathbb{H}^n$, for $n\geq 2$. For that purpose, the kissing number is replaced by the kissing function $κ(n, r)$ which depends on the radius $r$. After we obtain some theoretical lower and upper bounds for $κ(n, r)$, we study their asymptotic behaviour and show, in particular, that $\lim_{r\to \infty} \frac{\log κ(n,r)}{r} = n-1$. Finally, we compare them with the numeric upper bounds obtained by solving a suitable semidefinite program.