The Blue Cross-Blue Shield insurer is asking the court to review last week's rejection by a federal appeals court.
The appeals court upheld a federal judge's ruling that the deal would further reduce competition an already concentrated insurance market. The ruling was made after regulators sued to block the deal.
Even Cigna wants this deal to go away. The insurer has sued Anthem and is seeking billions of dollars in damages.
But Anthem says it hopes "1960s-era merger precedents" that the courts relied on for their decision can be updated to reflect "the modern understanding of economics and consumer benefit."
Anthem announced its Cigna bid in 2015. It has touted the deal as a way to help the companies negotiate better prices with pharmaceutical companies, hospitals and doctor groups. The company also has said the acquisition would help cut expenses and add more customers, which allows insurers to spread out the cost of investing in technology to manage and improve care.
But critics that include doctor and consumer groups aren't comfortable with giving an insurer the power and leverage that would come from such a huge acquisition. They have argued that this combination will lead to fewer choices for insurance shoppers.
Industry experts have suggested any consumer impact from the deal would take years to materialize and could lead to savings in some areas but higher costs elsewhere.
Earlier this year, a federal judge blocked another big insurance deal, Aetna (AET) roughly $34 billion acquisition of Medicare Advantage coverage provider Humana (HUM). Aetna then said it was abandoning its deal.