Anthem, which includes the Blue Cross, Blue Shield and Amerigroup brands, has raised its bid to bring about the move, which would be the latest step toward an expected consolidation among the biggest U.S. health-insurance companies.

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NEW YORK — After getting the cold shoulder, U.S. health insurer Anthem said it’s raising its offer to buy smaller rival Cigna for $54 billion, including debt.

Indianapolis-based Anthem said Saturday it’s proposing $184 per share, with about 31 percent of that in Anthem shares and the remainder in cash. The offer represents a premium to Cigna’s stockholders of 18 percent over Cigna’s closing stock price Friday. Anthem said the bid also represents a premium of more than 35.4 percent based on the closing price of Cigna’s shares May 28, when industry merger talks began.

The announcement comes as investors have been speculating about the possibility of a major acquisition in a sector where size is becoming increasingly critical. Last month there were reports that another rival, Humana, was exploring a sale of itself.

Some analysts predict the nation’s five biggest health-insurance carriers, which also include Aetna and UnitedHealth Group, will consolidate into three.

Anthem said it has been in talks with Cigna, based in Bloomfield, Conn., to explore a potential combination since August. Anthem said it made its proposal public because the companies have not been able to come to an agreement.

The company also said it has submitted four written proposals since early June and made previous offers of $174 per share and $178 per share, according to a letter, written by Anthem’s Chief Executive Joseph Swedish to Cigna’s board of directors, that it made public Saturday.

Anthem said a big stumbling block has been what role David Cordani, CEO of Cigna, is seeking to have at the combined companies.

In particular, Anthem said Cordani has demand that he lead the combined company. Swedish said he would be chief executive of the combined company for two years and Cordani would be his No. 2 as president and chief operating officer. After that time, Cordani would be a candidate for the top job, but with nothing guaranteed, according to Anthem.

“We were stunned that the Cigna board continues to insist on a guaranteed CEO position for Mr. Cordani over choosing to allow its stockholders to realize the significant premium being offered,” Swedish wrote in the letter to Cigna’s board.

Cigna’s spokesman Matthew Asensio declined to comment.

In its statement, Anthem said the combined company would be “an industry leader” with greater than $115 billion in annual revenue. It noted Anthem and Cigna together “would gain meaningful diversification,” covering about 53 million combined medical members and strong commercial, government, consumer and specialty franchises.

It also cited Anthem’s longstanding Blue Cross and Blue Shield brand in 14 states and Medicaid footprint via its Amerigroup brand in 19 states, including Washington.

The health-care industry is facing intense pressure to squeeze costs and find ways to capture opportunities arising from the Affordable Care Act. An ever-larger share of the companies’ business is tied to government programs and the health law’s exchanges, where cost-conscious individuals buy their own plans.

Getting bigger also could give insurers increased leverage in negotiating rates with hospitals, many of which have expanded through their own mergers.