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Some people say only those who are having Yagnopaveetam are eligible to recite Gayatri Mantra and
some people say there was no such limitation, any one can recite Gayatri Mantra.
Which one is correct ?

After initiation one becomes qualified to recite any Mantra not just the Gayatri Mantra. And, for Gayatri Mantra the initiation is called the Upanayana, where the investiture of the sacred thread is done.

Brahmanas, Kshatriyas, and Vaishyas are called the twice-born. Their
first births take place when they are delivered of their mother's womb
; their second, when they duly accept the Gayatri Mantra from their
preceptors (21).

Thus made twice-born, and free from all other faults, they become
entitled to study the Vedas, Smritis and Puranas (22).

Vyasa Smriti Chapter 1.

Also, see the following verses from Manu Smriti:

2.171. They call the teacher (the pupil’s) father because he gives the Veda; for nobody can perform a (sacred) rite before the investiture
with the girdle of Munga grass.

2.172. (He who has not been initiated) should not pronounce (any) Vedic text excepting (those required for) the performance of funeral
rites, since he is on a level with a Sudra before his birth from the
Veda.

So, what 172 says is that one should not recite any Vedic text before he is initiated. And, the Gayatri Mantra is the foremost of all Vedic Mantras.

Therefore, it's only after the initiation one should chant the Mantra. The 2nd opinion, that anyone can chant it, is certainly not true.