The last verse of the extant version of Manu Smriti says:
12.126. A twice-born man who recites these Institutes, revealed by Manu, will be always virtuous in conduct, and will reach whatever
condition he desires.
This hints that only twice born persons are qualified to recite/read Manu Smriti.
Vyasa Smriti, however, explicitly states that only Dvijas, who wear the sacred thread, are qualified to read any Smritis.
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
Thus made twice-born, and free from all other faults, they become entitled to study the Vedas, Smritis and Puranas (22).
(Chapter 1, Verses 21 and 22)