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Tokyo's Nikkei share average advanced 0.8%, the highest point since the middle of last week.
The US Standard & Poor's 500 E-mini futures added 0.3%, indicating a firmer open on Wall Street on Thursday after the S&P 500 index edged up 0.1% overnight. US Treasury futures eased 7-1/2 ticks, however.
House Democrats journeyed to the White House to discuss the stalemate on Wednesday, and Senate Democrats and Republican leaders in the House of Representatives will make separate visit on Thursday amid intensifying concerns about the potential for economic havoc.
Some Republicans and Democrats also floated the possibility of a short-term increase in the debt limit to allow time for broader negotiations on the budget.
Financial markets have been optimistic that a deal between the two sides would be reached by the October 17 deadline to avoid an historic US debt default.
"It's a step forward for the market to resume risk-taking, though we are not too optimistic," said Isao Kubo, an equity strategist at Japan's Nissay Asset Management. "Investors are cautiously buying back."
Still, strains in short-term interest rates and funding markets increased as the date drew ever closer, keeping investors on edge.
"We think a resolution to the debt ceiling impasse is increasingly likely to be a last minute affair, and market anxiety seems likely to build up as we head into next week. We expect to see dollar/yen test lower in the days ahead," analysts at BNP Paribas wrote in a note.
MSCI's broadest index of Asia-Pacific shares outside Japan slipped 0.4%, dragged lower by a 0.7% decline in China's CSI300 index.
The Asian gauge is flat on the year, though its 12-month forward price-to-earnings ratio slipped to 11.7, a level not seen in a month, according to Thomson Reuters Datastream.
BREATHING SPACE
The dollar was up 0.3% at 97.69, adding to a 0.5% rise in the previous session - its best one-day percentage rise in two weeks. It hit a five-week low against the Japanese currency on Tuesday.
The minutes of the Federal Reserve's September meeting revealing the decision not to reduce its $85 billion-a-month bond-buying programme was a "close call" also helped buoy the US currency.
"This is consistent with our expectations that the Fed will taper purchases at the upcoming December meeting. That said, the ongoing federal government shutdown and upcoming expiration of the debt ceiling suggests that the decision to taper could be pushed into 2014," Barclays Capital said in a note.
"A sooner resolution to the fiscal risks that cloud the outlook could keep December on the table, but a longer stalemate could dampen growth sufficiently and lead to a tapering in the first quarter of 2014 or later," it added.
Earlier on Wednesday, the dollar also gained on the news that Federal Reserve Vice Chairwoman Janet Yellen will be nominated as the next head of the US central bank.
Although the news could have been viewed as negative for the dollar given that investors regard Yellen as a policy dove, it helped soothe sentiment as the nomination was seen as reducing uncertainty in a market gripped by fear of a US debt default.
Against a basket of major currencies, the dollar gained 0.2% to 80.526, having climbed 0.4% on Wednesday to move further away from an eight-month trough.
The Australian dollar rose a third of a US cent after a better-than-expected jobs report prompted investors to pare back expectations of further easing by the country's central bank.
In the wake of the US fiscal concerns, Japanese investors sold a record amount of foreign bonds on a net basis last week, offloading nearly $23 billion worth.
In commodity markets, as the dollar regained some of its footing, gold eased 0.2% to around $1,304 an ounce, adding to Wednesday's 0.9% decline.
US crude prices steadied at about $101.7 a barrel, stabilising after a 1.9% tumble in the previous session as the largest weekly buildup of US stocks in a year weighed further on a market already concerned that Washington's stalemate would curb demand in the world's biggest oil consumer.
