Congress must agree by the end of February to increase the limit on how much the U.S. can borrow so the government can service its debt. If it doesn't, the country could default, which would deal a heavy blow to global financial markets and undermine confidence in the world's largest economy.
The Republicans appear ready to raise the debt ceiling temporarily and have also backed away from their insistence on deep spending concessions in exchange for a deal. The signs of compromise encouraged investors to buy into stock indexes, many of which are near multi-year highs.
"Although this again could be seen as another round of political battle, any progress to avoid immediate dangers will likely be seen as positive by the market," said Gary Yau, analyst at Credit Agricole CIB, in a report to investors.
Britain's FTSE 100 was up 0.3 percent to 6,171.71 while Germany's DAX advanced 0.5 percent to 7,736.55. France's CAC-40 gained 0.4 percent to 3,755.18.
U.S. stock and bond markets are closed for Martin Luther King, Jr. Day.
Dickie Wong, executive director of research at Kingston Securities in Hong Kong, said he was optimistic that an agreement on the U.S. debt ceiling would be reached because of the high price tag attached to failing to do so.
"Both parties will find some kind of solution because they all know that the debt ceiling will have to be increased," Wong said. "At the very last minute, they will sort it out."
Earlier in Asia, markets were more cautious, with Japanese shares hit hard by a rise in the yen. The Nikkei 225 fell 1.5 percent to close at 10,747.74.
The Bank of Japan began a two-day policy meeting and has been under pressure from the new government to take more aggressive steps to fight the long deflationary slump in the world's third largest economy. Some analysts expect the bank to expand its asset-purchasing program and set an inflation target.
South Korea's Kospi dropped 0.1 percent to 1,986.86. Hong Kong's Hang Seng fell 0.1 percent to 23,590.91. Australia's S&P/ASX 200 rose 0.1 percent to 4,777.50.
In mainland China, the Shanghai Composite Index gained 0.5 percent to 2,328.22 while the smaller Shenzhen Composite Index rose 0.7 percent to 942.50.
In commodity markets, the benchmark oil contract for February delivery was down 39 cents to $95.17 per barrel in electronic trading on the New York Mercantile Exchange. The contract rose 7 cents to finish at $95.56 per barrel on the Nymex on Friday.
In currencies, the euro fell to $1.3311 from $1.3320 late Friday in New York. The dollar fell to 89.67 yen from 90.03 yen.
Pamela Sampson in Bangkok contributed to this report.