"However, that may have only been the case because they were not immediately followed by a debt ceiling fight that held much higher stakes for investors," he wrote in a note. "If Washington does not kick the can by the end of September, and there is no eleventh hour resolution to the CR (continuing resolution), it may shake investors' confidence that the debt limit will be increased in mid-October. In contrast to the relatively mild consequences of government shutdowns, the debt ceiling has been a big threat to the stock market."
The House Friday approved a resolution to fund the government, but it simultaneously voted to defund Obamacare, which will never pass the Senate.
Barry Knapp, head of equity portfolio strategy at Barclays, said he is not worried about Congress shutting the government down or letting the U.S. hit the debt ceiling. But he does think the acrimony of the discussions could unsettle stocks temporarily. "We're not worried about the budget battle. We never were. We've made such improvement in the fiscal account. Government spending as a percent of GDP dropped four percent from the peak in 2009/2010 when it was up around 25.5 percent."
(Read more: Fed in 'monetary roach motel,' won't taper: Schiff)
Knapp said he expects Congress to compromise. "What we think is going to happen is they are going to get a short-term CR to get this to December, and then they're going to have a battle throughout the fall."
Traders said the debate in Congress is hanging over the market, but so is the fact that the economy is not perceived as strong enough to allow the Fed to pullback on some of its easing. The Dow fell 49 points Monday to 15,401, and the S&P 500 lost 8 to 1701. Treasury yields moved lower, with the 10-year yielding 2.71 percent.
Comments from New York Fed President William Dudley stoked concerns Monday morning when he said the Fed has yet to see "any meaningful pickup in the economy's forward momentum." While Dudley said the Fed could still move to slow down bond purchases this year, he noted that improvement in employment could be overstated. Dallas Fed President Richard Fisher took another tact in remarks later in the day.
(Read more: Fisher: Fed selection process 'terribly' mishandled)
Fisher criticized the Fed for not taking the first step to taper its $85 billion monthly bond purchases last week, as had been widely anticipated by Wall Street. Fisher said he had warned that doing nothing at last week's meeting would increase uncertainty about the future path of policy and "call the credibility of our communications into question."
Kansas City Fed President Esther George made a similar comment last week when she said the Fed created confusion in the market with its lack of decision to taper. George speaks again Tuesday, at 12:30 p.m. ET at the Chicago Fed payments conference. Cleveland Fed President Sandra Pianalto speaks at 8:30 a.m. ET at the same conference.
Knapp said the delay in the Fed's tapering led him to raise his year-end S&P 500 target to 1800 from 1600 last week.
"We just thought there was going to be a pullback, but we pushed that into the first quarter now," he said.
(Read more: DC's autumn chill means heated political battles)
"Our perspective on this is these Fed-related pullbacks are not terminal to the recovery," Knapp said. "They're not terminal to the bull market advance but they mark a point where investors take a pause."
Besides Fed speakers, investors will be watching home price data Tuesday. S&P/Case Shiller and FHFA home price data are released at 9 a.m. Consumer confidence for September is reported at 10 a.m.
Oil traders are also watching the UN where both President Barack Obama and President Hassan Rouhani speak Tuesday. The market (and world) are watching to see how the two leaders interact, after more than three decades of chilled relations.
—By CNBC's Patti Domm. Follow here on Twitter