I can’t say I know what basilica it might be. Here’s what I got from
newadvent.org:
A fragmentary account of a pilgrimage in the fourth century, preserved by Peter the Deacon (1037), mentions “a grotto at the place where the Jews the Savior captive”. According to the tradition it was in this grotto that Christ was wont to take refuge with his disciples to pass the night. It was also memorable for a supper and a washing of the feet which, according to the same tradition, took place there. Eutychius, Patriarch of Constantinople (d. 583), says in one of his sermons that the Church commemorates three suppers. “The first repast”, he says, “together with the purification, took place at Gethsemani on the Sabbath day, the first day, i.e. when Sunday was already begun. That is why we then celebrate the vigil” (P.G., LXXXVI, 2392). The second supper was that of Bethany, and the third was that was that of Holy Thursday at which was instituted the Holy Eucharist. Theodosius (c. 530) describes this grotto in these terms: “There [in the valley of Josaphat] is situated the basilica of Holy Mary, Mother of God, with her sepulchre. There is also the place where the Lord supped with his disciples. There he washed their feet. There are to be seen four benches where Our Lord reclined in the midst of His Apostles. Each bench can seat three persons. There also Judas betrayed the Saviour. Some persons, when they visit this spot, through devotion partake of some refreshment, but no meat. They light torches because the place is in a grotto”. Antonius of Plaisance (570), Arculf, Epiphanius the Hagiopolite, and others make mention of the well known pasch of which the grotto of Gethsemani was witness. In the Church of the Agony the stone was preserved on which, according to tradition, Jesus knelt during His Agony. It is related by the Arculf that, after the destruction of the church by the Persians, the stone was removed to the grotto and there venerated. In 1165 John of Wurzburg found it still preserved at this spot, and there is yet to be seen on the ceiling of the grotto an inscription concerning it. In the fourteenth century the pilgrims, led astray by the presence of the stone and the inscription, mistakenly called this sanctuary the Grotto of the Agony.