Step 1: Understanding the Concept:
While different Puranas glorify different deities as supreme (Vishnu in Vaishnava Puranas, Shiva in Shaiva Puranas), there is a strong philosophical tradition within the Puranic corpus that equates the Ultimate Reality (Brahman) with a specific deity who animates the universe. The term Vishvayatma (Vishva + Atman) literally means the 'Self or Soul of the World'.
Step 2: Detailed Explanation:
In the context of the Puranic and Upapuranic philosophical synthesis, especially in the Saura, Shaiva, and general Upapuranas, Shiva is often identified as the Vishvayatma. While Vishnu is the sustainer, Shiva in his transcendental form (Paramashiva) is viewed as the underlying consciousness that pervades all objects.
However, it is important to note that the term "Vishvayatma" is also a common epithet for the Sun God (Bhaskara) in the Vedas ("Surya Atma Jagatastasthushashcha"). In the Upapuranas specifically (like the Samba Purana and Saura Purana), the Sun is hailed as the soul of the universe.
But, looking at the standard Puranic theology that influenced the overall Upapuranic development, the identification of the Supreme Controller who dwells within everything as the 'Soul' is most prominently associated with Shiva in the larger Puranic corpus. Shiva is Sarvavyapi (all-pervading) and Antaryami (inner controller). In the Linga Purana and Maheshvara traditions, the universe is his body and he is the Atman. This specific question likely refers to this Saiva-influenced Vedantic synthesis common in the Upapuranas.
(Note: In sectarian Vaishnava Puranas, Vishnu would be the answer. But in general Puranic technical terminology, the 'Soul' as the underlying substratum is a characteristic often attributed to the Maheshvara/Shiva principle).
Step 4: Final Answer:
The deity is Shiva.