The statement you mentioned regarding mass increasing linearly with velocity is not accurate. In the theory of special relativity, the mass of an object does not increase with velocity. Instead, the concept of relativistic mass, which was used in older formulations, has been replaced by the concept of relativistic energy.
According to special relativity, as an object accelerates to speeds approaching the speed of light, its relativistic energy (which includes both its rest mass energy and its kinetic energy) increases. The energy of the object increases without bound as it approaches the speed of light, but its mass remains constant.
Now, regarding time dilation, it is not a result of the gravitational field generated by the mass of an object. Time dilation occurs due to the relative motion between two observers or due to the difference in gravitational potential between them. These effects are described by the theories of special and general relativity.
In the case of gravitational time dilation, which is associated with differences in gravitational potential, time passes more slowly in stronger gravitational fields. This means that an observer in a higher gravitational potential, closer to a massive object, experiences time passing more slowly compared to an observer in a lower gravitational potential. This effect is independent of the velocity of the objects involved.
In summary, time dilation can occur due to both relative motion (velocity) and differences in gravitational potential. The mass of an object does not directly affect time dilation, but it does contribute to the gravitational field, which can cause time dilation in its vicinity.