Sermons of disciples who are widely revered indicate that the Nirvana element is True Self.

Venerable Luang Phor Wat Paknam, Phra Monkol Thepmuni (Sodh Candasaro) of Wat Paknam, Bhasichareon, Bangkok, who practiced rightly according to the Four Foundations of Mindfulness and reached Dhammakaya gave the sermon “Tilakkhanatikatha” on August BE 2497 [CE 1954], it is taken from the book “Luang Phor Wat Paknum Biography and works, A compilation of 63 Sermons,” page 407.
As for the Arahant body, if you get there, it is truly permanent, happy and self. The Dhammakaya also has an aggregate, but it is Dhamma Aggregate. It is not the Five Aggregates. There is also an element, but a passionless element (Viragadhatu) and passionless dhamma (Viragadhamma). All of it is dhamma. Therefore, the Dhammakaya Arahant is totally permanent, happy and self. The refined Dhammakaya Arahant is also permanent, happy and self.
Venerable Phra Acharn Mun Bhuridattamahathera stated (Venerable Phra Acharn Mun Buridatta-mahathera, Muttodaya, Rungsaeng Publishing, P. 27):
Having all the treasures is not as sublime as having Self because Self is the place where all the treasures arise.
The Venerable gave a sermon on Nibbanadhamma as follows:
All of the Arahants have extinguished the three realms with shining success. Through examination and perseverance in practice, they have become perfectly trained (Bhavito bahuligato). This was a major accomplishment, to cultivate their minds to have the power to examine all conventional things and destroy them to become inactive (Akiriyà). Then, the Arahants were able to extinguish the three worlds. In the extinguishing of thethree worlds, all the Arahants did not fly up above the world of sensual desire, Pure Form or the Formless World. They remained where they were. Even Lord Buddha, the Venerable One just sat at the foot of the Bodhi tree. When about to extinguish the three worlds, Lord Buddha did not fly up above the three worlds, but extinguished them in the mind. The mind itself is where the three worlds exist. Therefore, those wishing to extinguish the three worlds must extinguish their own minds up to the destruction of action (Kiriya) and all that is unreal from the mind. There is only non-action [Nirodhadhatu] or [Nibbanadhatu] and Thiticitta and Thitidhamma [permenant mind and Dhamma], which never die [Amatadhamma].

Luang Pu Chop Thanasamo, Wat Pa Khokmon, Tambon Phanoi, Amphoe Wangsaphung, Loei Province states (Revised Biography, BE 2535, P. 129-130):
Nirvana is not annihilation, and it is not according to what the world reckons or speculates. If you truly practice correctly, you will see the Truth, know the Truth, see Nirvana yourself, see Lord Buddha, the Dhamma, the Sangha, and the teachers who are pure. All manner of doubt in you will cease.