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 about 40 narrations after collecting them from all previous shia books. But, how many of them are saheeh? Shaikh Asif muhsini in Mashraat Bihar ul Anwar vol-2, P-208, Ch: His birth and status of his mother, said: none of them is saheeh, all are zaeef & majhool.
This is what the top shia scholars have said about the birth of 12th Imam. Not a single report on his birth is saheeh from any of the shia books.
Moreover...
1. The reports are not in agreement specifying the day, month and year of his birth.
Shaikh Mufeed said in book 'Al-Fusul al-Mukhtarah' p.114: He was born on the 15th of Shaban 255H. On p.258 said: He was born in the year 252 A.H.
Sheikh Saduq, said in 'Ikmal al- Din': His birth was on 8th Shaban 256 A. H. Ref: 'Al- Fusul al-Mukhtarah' p.432.
Sheikh Tusi says in 'Al- Ghaybah'p.141 that: He was born in the 15th of Ramadan and in another report said he was born in the 15th of Shaban 255 A. H.
2. Reports also differed on his age at the time of the death of his father, between 2 and 8 years.
3. They also differed on the manner of pregnancy, whether in the womb or by the side.
4. They also differed on his complexion between being white or brownish.
5. The reports were inconsistent on the manner of his growth, being normal or fast.
6. They also diffred regarding the concealment of his affairs. i.e what happend to him after his father's death.
7. Some reports says that he was at fear of being arrasted, other say he was totally at peace.
8. Also Reports telling the story of his birth like report of hakeema, umairi etc all are weak, unknown & fabrications.
There is nothing in agreement regarding his existence. It looks like the story and charactor of 12th Imam was scripted by many individuals.
For more information refer this book Truth about 12th Imam (myth vs reality).docx