"I get a sense that investors want to rotate out of bonds into stocks and having a dovish Fed speaker helps," said Jack Ablin, chief investment officer at BMO Private Bank. "Having the Fed on the sidelines is probably the biggest thing—a Fed reluctant to tighten is emboldening investors."
Speaking early Monday at the Swedish capital, Chicago Fed President Charles Evans said the U.S. Federal Reserve could look at a rate hike in June if the economy is strong enough, Reuters said. His speech argued for rates to start rising in early 2016.
Evans told reporters if the Federal Open Market Committee had confidence that inflation was going to move up and that first quarter economic softness was temporary, "you could imagine a case being made for a rate increase in June."
Most of the second-quarter data reports have showed a slower-than-expected rebound from a weak first quarter.
"The focus is back on the macro," said Art Hogan, chief market strategist at Wunderlich Securities. "All things equal we're really going to have a focus on the European markets."
He noted little market-moving news on Monday for U.S. stocks as just a few retail names post earnings and most data reports come later in the week.
The only data out Monday was the National Association of Home Builders' survey, which showed builder confidence slipped two points in May, missing expectations of a slight gain. Housing starts and existing home sales come out later in the week, along with the Federal Open Market Committee's meeting minutes.
Apple closed up more than 1 percent after activist investor Carl Icahn said in an open letter to CEO Tim Cook that the tech company's shares remain undervalued and are worth almost double the current price.
A U.S. appeals court on Monday reversed part of the $930 million verdict that Apple won in 2012 against Samsung Electronics. In a mixed ruling, the court said the iPhone maker's trade dress (a trademark on how a product is presented for sale) could not be protected.
In Europe, stocks closed higher amid abating fears that Greece is on the verge of bankruptcy.
"Regardless of how the Greek impasse resolves itself, the global financial markets' ability to blissfully ignore Greece suggests that it expects one of two outcomes: (i) that a Greek default/exit is nowhere in the realm of possibility, or that (ii) even if Greece were to exit the Euro, the authorities have appropriately 'foamed down the runway' to curtail any contagion and the world will emerge largely unscathed from Grexit," MatlinPatterson's chief risk officer Ashwin Bulchandani said in a note.
Earlier, U.S. stocks opened mildly lower as European equities dipped on news that a Greece government spokesman said on Monday that authorities will pay public-sector wages and pensions in May, but needs a deal with creditors by the end of the month.
Read MoreWatch out: Greece's 'endgame' is in sight
Greece proposed to its international lenders that Europe's bailout fund pay back maturing Greek government bonds held by the European Central Bank as a way to overcome a funding crunch, Finance Minister Yanis Varoufakis said on Monday in a Reuters report. Athens could then pay the European Stability Mechanism (ESM), at a later date, Varoufakis told the annual assembly of the Greek Industrial Federation.
National officials in Athens also sent a letter to the International Monetary Fund that showed Greece came close to defaulting on a 750 million euro ($860 million) repayment last week, local newspaper Kathimerini and the Financial Times reported.
"Some of those markets in Europe have come under some pressure because of concerns about Greece and high flying bond yields," said Nick Raich, CEO of The Earnings Scout.
The DAX reversed losses to close higher, helped by a weaker euro at $1.135, while the ATHEX Composite ended up 1.6 percent after dipping 2 percent.
Greek bond yields soared, with the 10-year leaping 7 percent, as a leaked internal memo from the International Monetary Fund said the country had little chance of making the June 5 payment.
Spanish and Italian bond yields also briefly jumped more than 7 percent, while the German bund yield gained 3 percent to yield 0.65 percent.
U.S. Treasury yields edged higher, with the 10-year note yield briefly topping 2.23 percent and the 30-year climbing as high as 3.03 percent. Most analysts said bond yields remain in a range, as longer-term yields remain below 6-month highs touched last Tuesday.
"The rise in yields is not a concern about inflation or about the Fed. It simply reflects that global growth is better," Canally said.
The U.S. dollar gained more than 1 percent after posting five straight weeks of losses on Friday.
Analysts noted that morning losses in U.S. equities remained muted despite the renewed concerns about Greece.
"The market is acting on technical factors and remains strong," said Peter Cardillo, chief market economist at Rockwell Global Capital. "I think a deal will be reached (on Greece)—probably get down to the wire. Investors will keep an eye on that."
"We're probably going to stay range-bound with the S&P 500 trying to reach 2,130 - 2,135," he said.
Read MoreMarket's new biggest fear: Economic stall
The S&P 500 eked out a record on Friday for the second day in a row, while the other major indices advanced towards recent highs.
Crude oil pared early gains, with the contract for June delivery settling down 26 cents, or 0.44 percent, at $59.43 a barrel, as dollar strength and a bearish analyst outlook weighed on the commodity. Also adding to some volatility, the June contract will expire Tuesday after the close.
Earnings expected after the closing bell on Monday include Urban Outfitters, Agilent and Momo.