In the tangent space of the vector bundle $\xi=(E,\pi,M)$, $\dim M=n$, $\dim E=n+m$, using two different nonlinear connections $N^a_i(x,y)$ and $\bar N^a_i=N^a_i+T^a_i$ ($T^a_i$ is a tensor field), two adapted bases $B=\{\sigma_i,\partial_a\}$ and $\bar B=\{\sigma_i,\partial_a\}$ are introduced. The relations between the components of the generalized linear connection $\nabla$, the metric tensor and the torsion tensor expressed in these two bases are given. It is proved that the distinguished $d$-connection in the basis $B$ will be $d$-connection in the basis $\bar B$ iff $T^a_i$ is $h$- and $v$-parallel tensor field. The metrical connection $\nabla$ in the basis $B$ will be also metrical in the basis $\bar B$. The coefficients of the metrical generalized connections, when $T_H(E)$ is not orthogonal to $T_V(E)$, are obtained. They are functions of the given metric tensor, nonlinear connection and arbitrary torsion tensor. When the metrical generalized connection is compatible with co-symplectic, almost Hermitian, conformal, almost complex, almost product etc. structures, then, some restrictions for the torsion tensor are obtained, but their exact form is an open problem. For the special case, for the Finsler bundle and $d$-connection most of these problems are solved [l, 9, 13, 14].