Let $G$ be a graph without loops or bridges and $T_G(x,y)$ be its Tutte polynomial. We give sufficient conditions for the inequality $T_G(x,y)T_G(y,x)\geq T_G(z,z)^2$ to hold. We deduce in particular that $T_G(x,0)T_G(0,x)\geq T_G(z,z)^2$ for all positive real numbers $x,z$ with $x\geq z(z+2)$. Our result was inspired by a conjecture of Merino and Welsh that $\max\{T_G(2,0),T_G(0,2)\}\geq T_G(1,1)$.