House Speaker John Boehner and Senate Minority Leader Mitch McConnell said they had offered higher tax revenue as part of a deal. That could include limiting tax deductions for the highest earners.
Monday's gain for the Dow was its biggest since Sept. 6.
The S&P 500, meanwhile, rose above a key technical level, said Randy Frederick, managing director of active trading and derivatives at the brokerage Charles Schwab.
For nearly two weeks, the index has closed below its 200-day average, which on Monday stood at 1,382. It surpassed that marker Monday afternoon. Frederick said that might signal more buying.
Technical levels are historic averages and other indicators used by some traders to decide if stocks are a good value.
The market is closed on Thursday for Thanksgiving and will close early Friday.
On Monday, stock indexes in France, Germany and Britain closed up 2.5 percent as traders hoped that Greece would get its latest round of bailout cash.
Greece needs international lenders and the International Monetary Fund to release the money so that Greece can meet upcoming payments to creditors. Trading in Europe is still volatile. The region has entered recession.
Finance ministers from nations that use the euro will meet Tuesday. Later in the week, leaders will convene to discuss the European Union's budget for the next few years.
Benchmark crude oil rose $2.36, or 2.7 percent, to finish at $89.28 per barrel in New York. Conflict between Israel and Hamas pushed the price up, as did hope for a fiscal pact, which would help the U.S. economy and raise demand.
Earlier, Asian markets rose more modestly.
The yield on the benchmark 10-year Treasury note rose to 1.61 percent from 1.58 percent late Friday, a sign that traders are selling low-risk investments. A bond's yield rises as its price falls.
The market's longer-term direction will likely hinge on U.S. leaders' ability to attack the fiscal challenge between Thanksgiving and Christmas, Frederick said.
"If they can put some sort of a plan together, or make us believe they have a plan, or at least that there's some cooperation going on there, that could be a real boost for the market," he said.
Among big companies making news, Intel's CEO of seven years announced that he will retire in May. The stock rose 6 cents to $20.25.
Diamond Foods hit its lowest price since September 2006 after an analyst cut the snack food company's rating and price target. Diamond restated two years' worth of financial results last Wednesday, effectively wiping $56.5 million in profit from its books. Diamond fell $1.79, or 11.8 percent, to $13.34.
Advanced Micro Devices Inc. rose after an analyst said that the stock could see an "early 2013 bounce." He said investor concerns about the company's solvency were overblown and caused traders to oversell. AMD rose 6 cents, or 3.2 percent, to $1.92.
Daniel Wagner can be reached at www.twitter.com/wagnerreports .