Starting from the notion of conformal metrical structure in the tangent bundle, given by R. Miron and M. Anastasiei in [10, 11], we define the notion of conformal metrical $d$-linear connection with respect to a conformal metrical structure corresponding to the 1-forms $\omega$ and $\tilde\omega$ in $TM$. We determine all conformal metrical $d$-linear connections in the case when the nonlinear connection is arbitrary and we give important particular cases. Further, we find the transformation group of these connections. We study the role of the torsion tensor fields $T$ and $S$ in this theory, especially the semi-symmetric $d$-linear connections, and the group of transformations of semi-symmetric conformal metrical $d$-linear connections, having the same nonlinear connection $N$ and its important invariants.