The authors consider the integral transform called index transform \[ g(x)= \int_{-\infty}^\infty \tau F_{i\tau}^\varphi(x) f(\tau) d\tau, \quad x> 0, \] where the function \(f\) belongs to the Lebesgue space \(L^p(\mathbb{R})\), \(1\leq p<+\infty\), and the kernel \(F_{i\tau}^\varphi(x)\) is given by \[ F_{i\tau}^\varphi(x)= \frac{1}{4\pi i} \int_{\gamma-i\infty}^{\gamma+ i\infty} \Gamma\biggl( s+\frac{i\tau}{2} \biggr) \Gamma\biggl( s-\frac{i\tau}{2} \biggr) \varphi(s) x^{-s} ds, \] with \(\gamma>0\) and \(\varphi\) a measurable function such that this integral is convergent. The index transform comprises the Kontorovich-Lebedev and Mehler-Fock transforms. Using the relation \[ F_{i\tau}^\varphi(x)= \int_0^\infty K_{i\tau} (2\sqrt{t}) \Phi\biggl( \frac{x}{t} \biggr) \frac{dt}{t}, \] where \(K_{i\tau}\) is the Macdonald function of index \(i\tau\), and \(\Phi(y)= \frac{1}{2i\pi} \int_{\gamma-i\infty}^{\gamma+ i\infty} \varphi(x) x^{-s} ds\), and the condition that \(\Phi\) belongs to \(L_{\nu,r} (\mathbb{R}_+)\), \(\nu>0\), \(r\geq 1\) (the space of measurable functions on \(\mathbb{R}_+\) such that \(\int_0^\infty t^{\nu r-1}| f(t)|^r dt<+ \infty)\), they prove that the image of the index transform belongs to \(L_{\nu,r} (\mathbb{R}_+)\) and they give inversion formulas for this transform.