Recall the commutative story: for a commutative algebra $A$, its module of differentials $\Omega (A)$ is characterized by the universal property that any derivation $\delta \colon A \to M$ is in a unique way a composition $A \to \Omega(A) \to M$, where $d \colon A \to \Omega (A)$ is the canonical derivation.
(In what follows, I will be careless with signs and shifts.) Now let $A$ be a noncommutative DG-algebra over a field $k$, $M$ a DG-bimodule over $A$. Consider the following version of Hochschild cochain complex: $$C^n(A,M) = \bigoplus_{i+j=n}\operatorname{Hom}_k^i(A^{\otimes_k j},M).$$
The differential $d_H$ consists of two parts, where for $f \in \operatorname{Hom}_k^i(A^{\otimes_k j},M)$, $$d_1(f)(a_1\otimes \ldots \otimes a_j) = d_mf(a_1 \otimes \ldots \otimes a_j) + \sum \pm f(d_{A^{\otimes_k j}} (a_1 \otimes \ldots \otimes a_j));$$ $$d_2(f)(a_0 \otimes \ldots \otimes a_n) = a_0f(a_1 \otimes \ldots \otimes a_n)+\sum_i \pm f(a_0 \otimes a_i a_{i+1} \otimes \ldots a_n) +\pm f(a_0 \otimes \ldots \otimes a_{n-1})a_n.$$
One can see that for $f \in Hom_k(A,M)$, the equation $d_H(f)=0$ means that $f$ commutes with differentials and $f(ab) = af(b)+f(a)b$. That is, f is a biderivation.
More general closed elements of this complex can interpreted as "homotopy biderivations". Consider $f \in C(A,M)$, where $f = f_0 + f_1 + f_2+ \ldots + f_n$, with $f_i \in Hom^{-i}(A^{\otimes i},M)$. Then $d_H(f) = 0$ says that: 1) $f_0$ is closed, 2) $f_1$ doesn't respect the differentials, but the commutator is the commutator with $f_0$, 3) $f_1$ is a biderivation up to homotopy given by $f_2$, etc. Let's call such $f$ a homotopy biderivation.
Is there a version of universal DG-bimodule for homotopy biderivations, along the lines of $\Omega(A)$ in the commutative story? I would be grateful for any relevant links.