The world of American health insurance may soon become even smaller, with the biggest companies seeking to become even bigger.

A scramble has broken out within the industry as various providers jockey for position and make overtures to rivals. Anthem made the first public move, unveiling a $47 billion takeover bid for Cigna on Saturday after months of negotiations had stalled. On Sunday, Cigna fired back, rejecting the bid as “inadequate and not in the best interest of Cigna’s shareholders.”

But others have been quietly maneuvering as well. UnitedHealth Group, the biggest American health insurer by revenue, recently made a preliminary approach to Aetna, a person briefed on the matter said.

And a number of companies have indicated their interest in buying Humana, one of the smaller major insurers but one with a valuable Medicare franchise. Among those companies that had expressed interest is Anthem, though the bigger insurance provider is currently focused on combining with Cigna, people briefed on the company’s plans said. Another is Cigna.

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Jeffrey R. Ungvary President