NEW YORK (TheStreet) -- Major U.S. stock markets jumped Friday following yesterday's second-biggest gains of the year as signs of constructive discussions in Washington bolstered confidence the government won't allow the U.S. to default on its debt obligations.

The S&P 500 gained 0.63% to 1,703.19 after the benchmark's biggest jump since Jan. 2, while the Dow Jones Industrial Average added 0.73% to 15,237.11. The Nasdaq closed up 0.83% to 3,791.87. 

House Republicans offered to end the government nine-day-old government shutdown in exchange for budget cuts, according to reports from the Associated Press. Republican leaders have been pressing President Obama to cut funding for the Affordable Care and Patient Protection Act before they would agree to raise the limit on federal borrowing to pay for government expenditures and programs previously approved by Congress.

In equity trading, Safeway (SWY) was the most prominent gainer in the broad market as shares jumped 6.9% to $33.75 following the supermarket operator's announcement that it will by early 2014 exit the Chicago market where it operates 72 Dominick's stores and that the move will result in a cash tax benefit of $400 million to $450 million. Excitement swirling the announcement was helping to distract from its third-quarter misses.

Oil refiners such as Tesoro (TSO) and Valero Energy (VLO) - Get Report also gained ground. Tesoro popped 3.8% to $45.19 while Valero jumped 2.7% to $36.53 supported by easing concerns about an economic fallout from the Washington debates and the impact on demand for refined petroleum products, as well as a proposal by the U.S. Environmental Protection Agency proposal to ease its ethanol mandate next year to a requirement that 15.21 billion gallons of ethanol and biodiesel be blended into motor fuels. That's down from the current 18.15 billion gallon requirement and would substantially reduce costs for oil refiners.

The benchmark 10-year Treasury was little changed, yielding 2.687%, while the one-month bill was climbing 1/32, diluting the yield to 0.261% after popping for much of the morning.

On Thursday evening, during a 90 minute discussion, House Republicans proposed a temporary, six-week increase to the debt limit so that the U.S. may keep borrowing money to pay its bills while a bigger deficit reduction deal was discussed. The president said he was open to further discussions and also urged for an immediate end to the government shutdown. A statement issued shortly before the market closed from House Speaker John Boehner's spokesman said the president and he agreed they should keep talking.

The Reuters/University of Michigan's consumer sentiment index indicated a drop to 75.2 in the preliminary October reading from 77.5 in September versus economists' average expectation of 76, reflecting the psychological impact of the tension in Washington as the government shutdown heads into its 11th day.

The S&P 500 rebounded from losses in the first half of the week, closing higher by 0.75%. The Dow gained 1.09% for the week, while the Nasdaq lost 0.42% for the five-day trading period.

-- Written by Andrea Tse and Joe Deaux in New York

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