According to our current understanding of physics, as an object accelerates towards the speed of light, its mass does not increase. This idea was proposed by Albert Einstein in his theory of relativity.
In Einstein's theory, mass and energy are related through the famous equation E=mc², where E is the energy, m is the mass, and c is the speed of light in a vacuum. This equation shows that energy and mass are interchangeable and that mass can be thought of as a form of stored energy.
When an object accelerates, it gains kinetic energy, which increases its total energy. However, this increase in energy does not cause an increase in mass. Instead, the object's relativistic mass, a term used in older theories, appears to increase, but it is now more accurately understood as an increase in the object's total energy.
As an object approaches the speed of light, its energy increases without limit, but its mass (in the modern understanding) remains constant. This is why we say that the mass of an object does not increase as it approaches the speed of light.