Gr\"{u}ss showed that if $f$ and $g$ are integrable real functions on $[a, b]$ and there exist real constants $\alpha,\beta,\gamma,\Gamma$ such that $\alpha\leq f (x)\leq \beta$ and $\gamma\leq g(x)\leq \Gamma$ for all $x\in [a, b]$, then \begin{eqnarray*} \left|\dfrac{1}{b-a}\int_a^b f(x)g(x)dx-\dfrac{1}{b-a}\int_a^b f(x)dx\ \dfrac{1}{b-a}\int_a^b g(x)dx\right|\leq \frac{1}{4}|\beta-\alpha| |\Gamma -\gamma|. \end{eqnarray*} This inequality was studied and extended by a number of mathematicians for different contents such as inner product spaces, quadrature formulae, finite Fourier transforms and linear functionals. For unital $n$-positive linear maps $\Phi$ ($n\geq 3$), the authors of [3] proved that \begin{eqnarray*} \|\Phi(AB)-\Phi(A) \Phi(B) \|\leq \inf_{\alpha \in \mathbb{C}}\|A-\alpha I\| \inf_{\beta \in \mathbb{C}}\|B-\beta I\|. \end{eqnarray*} for all operators $A,B$ in a $C^*$-algebra. The Gr\"{u}ss inequality was generalized in the setting of inner product modules over $H^*$-algebras and $C^*$-algebras in [1]. Several Gr\"{u}ss type inequalities in inner product modules over $C^*$-algebras are investigated in [2]. In this talk, we investigate several new Gr\"{u}ss type inequalities for positive linear maps

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