We study a new class of integral transforms from $L_p(\mathbb R_+)$ to $L_q(\mathbb R_+)$, $1 \leq p \leq 2$, $p^{−1} + q^{−1} = 1$, related to the Fourier cosine convolution with a weight function. We obtain necessary and sufficient conditions under which the new transforms are unitary in $L_2(\mathbb R_+)$. A Plancherel type theorem and the boundedness of these integral operators are obtained. We also give several examples of the new transforms kernels.