Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)?

More precisely, let $$ f = a_0 + a_1 X + \cdots + a_{n-1}X^{n-1} + X^n $$ with $a_i \in \mathbb{Z}$ and $\lvert a_i \rvert < M$. Suppose $f(z)=0$ implies $\lvert z \rvert > 1$ (so all zeros have absolute value greater than $1$).

What is an (explicit) positive function $B(n,M)$ with $$\lvert z \rvert - 1 \geq B(n,M)$$ for any zero $z$ of $f$?

If it is easier, then $M=2^n$ can be assumed. Further, I am only interested in the behavior for large $n$, i.e., I want something like $$ \frac{1}{\lvert z \rvert - 1} = O(B(n,2^n)) $$ for $n\to\infty$ and for each zero $z$ of any monic polynomial $f$ with degree at most $n$ and integer coefficients bounded by $2^n$.