Health insurers say they plan to raise premiums for some Americans as a direct result of the health overhaul in coming weeks, complicating Democrats' efforts to trumpet their signature achievement before the midterm elections.

Aetna Inc., some BlueCross BlueShield plans and other smaller carriers have asked for premium increases of between 1% and 9% to pay for extra benefits required under the law, according to filings with state regulators.

These...