The offer was made privately on Friday. Although the White House rejected the majority of Boehner's proposal, Democrats said the offer was a potential opening for further budget talks and an eventual compromise, The Wall Street Journal reported.
Gains were held in check by a U.S. Federal Reserve of New York report that pegged New York state manufacturing as sliding further in December, although the pace of decline was described as modest.
By close of trading, the Dow Jones industrial average added 100.38 points, 0.76 percent, to 13,235.39. The Standard and Poor's 500 gained 16.78 points, 1.19 percent, to 1,430.36. The tech-heavy Nasdaq composite added 39.27 points, or 1.32 percent, to 3,010.60.
On the New York Stock Exchange, 2,053 stocks advanced and 1,007 declined on a volume of 3.4 billion shares traded.
In Tokyo, the Nikkei 225 index added 91.32 points, 0.94 percent, to 9,828.88.
In London, the FTSE 100 index lost 0.16 percent, 9.61 points, to 5,912.15.
The benchmark 10-year U.S. Treasury note fell 20/32 to yield 1.777 percent.
The dollar rose to 83.88 yen, up from Friday's 83.66 yen. The euro rose to $1.316 from Friday's $1.3075.