The Dow Jones industrial average ended 130 points higher, with Boeing, Goldman Sachs and 3M contributing the most gains. At session highs, the index traded 271.20 points higher.
"We're basically putting on what Brexit took off last week," said Mark Luschini, chief investment strategist at Janney Montgomery Scott. "I think we're going to be held hostage to [Thursday's] vote."
"I think we're going to continue to see some nervousness [in the market]," said JJ Kinahan, chief strategist at TD Ameritrade.
The Nasdaq composite outperformed. The iShares Nasdaq Biotechnology ETF (IBB) closed about 1 percent higher, after briefly trading more than 1.5 percent higher.
U.S. futures surged Monday, with Dow futures advancing more than 200 points, while S&P and Nasdaq futures had gained about 25 and 50 points, respectively.
"The polls are shifting in Great Britain, and so are the markets," said Peter Cardillo, chief market economist at First Standard Financial, adding investors are also covering short positions. "Last week you had Brexit worries, options expirations, the Fed, and that was a combination that weighed on the market."
Major global indexes also rose. The pan-European Stoxx 600 index gained more than 3.5 percent with banks outperforming, while Asian equities climbed overnight.
"Equities may continue to rally if the vote is an official Remain on Thursday but sovereign bonds may be where the real action could be considering how stretched that rubber band is," said Peter Boockvar, chief market analyst at The Lindsey Group.
U.S. Treasury yields rose, with the 2-year yield around 0.73 percent and the 10-year yield was 1.67 percent as of 4 p.m. ET.
The 10-year German bund yield held in positive territory after falling into negative territory last week for the first time.
The Treasury auctioned $26 billion in two-year notes at a high yield of 0.745 percent on Monday. The bid-to-cover ratio, an indicator of demand, was 2.72.
The U.S. dollar index traded more than half a percent lower, with the euro near $1.131 and the yen near 103.9 yen versus the dollar. The pound surged more than 2 percent against the dollar to trade near $1.467 and on track for its best one-day performance in seven years, according to Reuters.
"The UK's future is much better within the EU, something we have said several times before and we maintain our view because leaving the EU may only create problems for the country," Naeem Aslam, chief market analyst at Think Forex, said Monday in a note to clients.
"Having said that, we need a firm confirmation that public opinion has indeed changed and is leaning more towards the 'Remain' side before we can see the current momentum continue," he said.
The CBOE Volatility Index (VIX), widely considered the best gauge of fear in the market, fell sharply to trade near 18.
"Some people had bought some protection into the weekend," TD Ameritrade's Kinahan said. " "I think that might have been a stopgap for the weekend."
Neel Kashkari, president of the Federal Reserve Bank of Minneapolis, said Monday in a Reuters report that regulators may hurt the economy and push borrowers toward non-bank lenders if they set too-high capital standards for Wall Street.
He told reporters that Brexit could have "moderate direct effects" on the U.S. economy but probably does not pose big financial risks.
On Tuesday, Fed Chair Janet Yellen is scheduled to testify in front of Congress on a number of matters, including monetary policy.
"She's pretty consistent; I don't think she'll stray too far from what she's said recently," Fort Pitt Capital's Forrest said.