Is it permissible to enter into the marriage contract with a woman when she has her monthly period?.

Praise be to Allaah.  

The basic principle concerning that is that it is permissible. There is nothing to suggest that it is not allowed in the Qur’aan, Sunnah, scholarly consensus, words of the Sahaabah or correct analogy. I do not know of any scholar who regarded that as haraam or makrooh. But some of the fuqaha’ regarded it as makrooh for a woman to have the wedding party if she is menstruating, lest her husband have intercourse with her at that time and thus fall into sin. 

The common folk confuse the ruling on this matter with the ruling on divorcing a woman when she is menstruating; there is nothing in common between the two. 

It is permissible to marry a woman when she is menstruating, according to consensus, and it is haraam to divorce a menstruating woman with whom one has had intercourse, according to consensus.