Step 1: Understand viral genetic material.
Viruses may contain either DNA or RNA as their genetic material. RNA viruses can further possess single-stranded RNA (ssRNA) or double-stranded RNA (dsRNA).
Step 2: Analyze Influenza virus.
Influenza virus contains single-stranded negative-sense RNA as its genetic material. Therefore, it is not a double-stranded RNA virus. Hence, option (A) is incorrect.
Step 3: Analyze Rous sarcoma virus.
Rous sarcoma virus is a retrovirus containing single-stranded positive-sense RNA. It uses reverse transcriptase during replication. Thus, option (B) is incorrect.
Step 4: Analyze Polio virus.
Polio virus belongs to the Picornavirus group and contains single-stranded positive-sense RNA. Therefore, option (C) is incorrect.
Step 5: Analyze Reovirus.
Reoviruses are well known for possessing segmented double-stranded RNA genomes. Hence, option (D) is correct.
Step 6: Compare all the options.
Among the given viruses, only Reovirus contains double-stranded RNA as its genetic material.
Step 7: Final conclusion.
Therefore, the animal virus containing double-stranded RNA is Reovirus.
\[
\boxed{\text{Reovirus}}
\]
Hence, the correct answer is option (D).