Sir, as sai is one of the important rituals of Haj and Umra, what is its background in terms of Islam. According to one version ascribed to a surah of holly Qur’an, it is reminiscent of tawaf performed by Hazrat Ibrahim and sacrifice of Hazrat Ismail after sighting him at marwa from safa hill, while according to another tradition it signifies the making of hectic rounds between the two hills in search of water for crying infant Hazrat Ismail by Hazrat Hajira which led to the emergence of spring water named as aabe zam zam.
It is difficult to establish what exactly the Sai refers to. As you correctly wrote it can be referring to Ibrahim (as) sacrificing Ishmael (ra) or it can be referring to Hajar wondering around the two mountains, searching for water for the infant Ishmael.
- All of the obligatory worships of Hajj relate to Ibrahim (as). It is unlikely that this particular one would not relate to him (directly).
- We are certain from the Qur’an that Ibrahim (as) attempted to sacrifice Ishmael (as). We are however not certain whether Hajar really ran between the two mountains in search of water (unless we rely on biblical sources for this).