Trader sentiment was also weighed by data showing China's economic recovery remained sticky, with key indicators missing expectations owing to weak domestic demand.
While there is a general feeling that an agreement will be reached, Republican House Speaker Kevin McCarthy warned Monday that staff-level meetings were "not productive at all" and they were "nowhere near reaching a conclusion".
For their part, Republicans are demanding spending cuts as a condition for passing the bill while Democrats want a "clean" increase of the borrowing limit with no strings attached.
US President Joe Biden, who has expressed confidence the two sides can bridge the gap, is scheduled to meet with McCarthy and other congressional leaders at the White House later Tuesday.
McCarthy's comments came after Treasury Secretary Janet Yellen again said the government would likely run out of cash on June 1, meaning it would not be able to meet its debt repayment obligations, sparking a potentially devastating default.
Meanwhile, two top Federal Reserve officials suggested they were in favour of pausing the US central bank's interest rate-hiking drive next month.
Chicago Fed boss Austan Goolsbee said he wanted to wait for the effects of more than a year of increases aimed at bringing inflation down from multi-decade highs.
"There is still a lot of the impact of the 500 basis points we did in the last year that's still to come," he told CNBC. "And you add on that there are tight credit conditions, and I think that we should be extra mindful.
"We need to take that into account, and the only way to do that is sit and watch it."
Atlanta Fed president Raphael Bostic added that he favoured staying put at the June meeting, though he threw cold water on any hope for a cut before the end of the year.
While Bostic made clear he favours putting the policy on hold for now, he also suggested that the next move may be more likely up than down, given the persistence of price pressures.
But Minneapolis Fed President Neel Kashkari said monetary policymakers still had plenty of work to do to rein in prices, citing the still strong labour market and the fact that inflation, at five percent, remained well above the bank's two percent target.
Below-forecast China data
Hong Kong dipped Tuesday despite a rally in tech firms following news that US investor Michael Burry -- who made his name predicting the 2008 housing crisis -- had boosted his investments in e-commerce giants Alibaba and JD.com.
However, below-forecast readings on Chinese retail sales, industrial production and fixed asset investment reinforced the view that the world's number two economy was still struggling to bounce back from years of tough zero-Covid measures.
"It is well understood that China's recovery will not in any way, shape or form (recover) linearly like the recoveries of yesteryears, especially with youth unemployment hitting 20 percent," said SPI Asset Management's Stephen Innes.
"That is an unsettling and scary number."
There were also losses in Shanghai, Sydney, Singapore, Mumbai, Jakarta and Bangkok, though Tokyo, Seoul, Taipei and Manila edged up.
London rose at the open though Frankfurt and Paris edged down.
Key figures around 0715 GMT
Tokyo - Nikkei 225: UP 0.7 percent at 29,842.99 (close)
Hong Kong - Hang Seng Index: DOWN 0.2 percent at 19,926.73
Shanghai - Composite: DOWN 0.6 percent at 3,290.99 (close)
London - FTSE 100: UP 0.1 percent at 7,781.96
Euro/dollar: DOWN at $1.0874 from $1.0878 on Monday
Pound/dollar: DOWN at $1.2480 from $1.2528
Dollar/yen: DOWN at 135.87 yen from 136.10 yen
Euro/pound: UP at 87.13 pence from 86.80 pence
West Texas Intermediate: DOWN 0.2 percent at $71.00 per barrel
Brent North Sea crude: DOWN 0.2 percent at $75.10 per barrel
New York - Dow: UP 0.1 percent at 33,348.60 (close)