An object vibrates near its resonant frequency. Will it change to its resonant frequency? That is to say, will he slowly become resonant?

I think you may want to ask: When an object vibrates near its resonant frequency (when the excitation source stops working), will it approach its resonant frequency? That is to say, will it slowly become vibrating at its natural frequency?

Let's talk about the theory first. Let's assume that the linear system of an object, when excited by a pulse signal, generates a damping vibration wave with a certain frequency (the amplitude gradually decreases). Any input signal can be decomposed into the sum of pulse signals, and the vibration of the object is the sum of these pulse signal responses, which is the definition of convolution.

then, when an object is forced to vibrate, the vibration frequency of the object is the frequency of the excitation source, and this vibration is actually the sum of countless impulse responses. Whether there is an excitation source or not, an object actually vibrates at its own natural frequency. When the excitation source stops working, the vibration of the object is still the superposition of multiple damping waves. It is hard to say that it will gradually change to vibrate at its natural frequency, but of course, the existence of special circumstances cannot be ruled out.