Aetna Inc. is negotiating with several states to move its corporate headquarters out of Connecticut, dealing a blow to a state that is struggling with a widening budget deficit, stagnant population growth and rising unemployment.

The health insurer said Wednesday that in uprooting from Hartford—its home since 1853—it aimed to broaden access to innovation and talent. The company has explored moving to Boston and New York, according to people familiar with the discussions.

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