Yes, it is known exactly what these verses are - they are known by the illustrious name Jaya, meaning victory. The Jaya is the core part of the Mahabharata, an 8800 verse poem within the larger epic consisting of the famous conversations between Dhritarashtra and his charioteer Sanjaya about the Pandavas' victory over the Kauravas in Kurukshetra. Sanjaya had been gifted by Vyasa with divine vision, so that he could narrate the events of the Mahabharata war to Dhritashtra. The Jaya begins in the Bhagavad GitaBhagavadgita Parva of the Bhishma Parva of the Mahabharata, when Sanjaya tells Dhritashtra that Bhishma has been defeated:
Possessing a knowledge of the past, the present and the future, and seeing all things as if present before his eyes, the learned son of Gavalgana, O Bharata, coming quickly from the field of battle, and rushing with grief (into the court) represented unto Dhritarashtra who was plunged in thought that Bhishma the grandsire of the Bharatas had been slain.
And it ends 8800 verses later in the Sauptika Parva of the Mahabharata, when Sanjaya tells Dhritarashtra how he lost his divine vision after the death of Duryodhana:
[Sanjaya said:] "Casting off his griefs for all his (slain) kinsmen, he then gave up his life-breath. His soul ascended to sacred heaven, while his body only remained on earth. Even thus, O king, thy son Duryodhana breathed his last. Having provoked the battle first, he was slain by his foes at last. The three heroes repeatedly embraced the king and gazed steadfastly on him. They then ascended their cars. Having heard these piteous lamentations of Drona's son, I came away at early dawn towards the city. Even thus the armies of the Kurus and Pandavas have been destroyed. Great and terrible have been that carnage, O king, caused by thy evil policy. After thy son had ascended to heaven, I became afflicted with grief and the spiritual sight which the rishi gave hath been lost by me!" ... The king, hearing of his son's death, breathed long and hot sighs, and became plunged in great anxiety.
Note that Sanjaya's words were divinely inspired, so even he may not have fully remembered what he had said to Dhritarashtra after the fact, which is why it says "I am (continued Sauti) acquainted with eight thousand and eight hundred verses, and so is Suka, and perhaps Sanjaya." (Shuka and Ugrashrava Sauti learned it formally, whereas Sanjaya was just recollecting his experience without trying to memorize it.)
In any case, it was after the composition of this Jaya that Vyasa went to Ganesha to have his epic written down, and that is when the poem grew from the 8800-verse Jaya to the 24000-verse Bharata. The Bharata is the version of the poem that Vyasa's son Shuka told to Arjuna's grandson Parikshit. Some time after his dictation to Ganesha, Vyasa expanded the poem even more, to the 100,000-verse epic poem we now call the Mahabharata. (And it is said that he composed several hundred thousand more verses which he revealed only to the gods and other beings, perhaps dealing with the unseen events going on in other lokas while the Mahabharata war was going on, akin to what was happening in the Battle of the Ten Kings.) This was the version that Vaishampayana told Parikshit's son Janamejaya, and the version that Ugrashrava Sauti told to the sage Shaunaka in the Naimisharanya forest.