The kinetic energy of a body of mass $m$ and velocity $v$ is given by $K=\frac{1}{2}mv^2$. Since the bodies have the same linear momentum, we can write:

$$p=mv$$

where $p$ is the linear momentum of the bodies.

Let the masses of the two bodies be $m_1$ and $m_2$ and their kinetic energies be $K_1$ and $K_2$, respectively. Then, we have:

$$\frac{K_1}{K_2}=\frac{16}{9}$$

$$\frac{1}{2}m_1v_1^2\div\frac{1}{2}m_2v_2^2=\frac{16}{9}$$

Since $p=mv$, we have $v_1=\frac{p}{m_1}$ and $v_2=\frac{p}{m_2}$. Substituting these in the above equation, we get:

$$\frac{m_2}{m_1}=\frac{9}{16}$$

Therefore, the ratio of the masses of the two bodies is $\boxed{9:16}$.