Kulayni in his book Usool e Kafi vol-2, Ch#125 Birth of Sahib, collected 31 narrations. As per research of Allama Majlisi in Mirat ul Uqul vol-6, P-170 onwards: Out of those 31 narrations only 8 (No.4,8,15,20,24,25,29,31) are saheeh, rest all are weak/unknown and unreliable. But, none of those 8 narrations mention anything about birth. They talk about other issues. According to Allama Behbudi in Sahi al-Kafi, Ch#1, Birth of Sahib, only 2 narrations are saheeh, 9th & 24th. In both these narrations there is no words on birth.
Allama Majlisi in Bihar ul Anwar vol-51, P-2 to 28, compiled