WASHINGTON—The government shutdown enters its second week with the two parties still bitterly divided and Republicans increasingly tying the fight to a fast-approaching deadline to avoid a default on U.S. debt.

House Speaker John Boehner (R., Ohio) said Sunday he wouldn't bring up bills to fully reopen the government or increase the country's borrowing limit unless Democrats agree to broader talks aimed at trimming the deficit. The speaker insisted he couldn't muster enough votes to pass either one without the concessions.

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