If y-axis is the approaching axis between two atoms, then which of the set of orbitals cannot form the $\mathrm{\pi }$-bond between two atoms in general?

Bonds formed by a side-to-side overlap of two orbitals are referred to as $\mathrm{\pi }$-bonds. Therefore, orbitals that reside in the Y-axis will only produce head-on overlaps, which are technically known as the $\mathrm{\sigma }$-bonds, if the Y-axis is the axis that connects the two atoms. Simply put, this means that there won't be any sideways overlap between orbitals in that axis and that there hasn't been any $\mathrm{\pi }$-bond creation up until this point.