According to U.S. media, the deal currently under discussion would fund the government until 15 January while raising the debt ceiling until early to mid-February.
Senate leaders have expressed optimism that a deal could be struck but any deal would also need to pass the House of Representatives.
Senate Democratic Majority Leader Harry Reid told the chamber on Monday: "We've made tremendous progress... perhaps tomorrow will be a bright day."
Tokyo stocks closed up 0.26 percent on the news.
The U.S. must raise its $16.7 trillion borrowing limit by Thursday.
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