David Kelly, chief global strategist at JPMorgan Funds, said investors were watching rather than reacting significantly to the Greek headlines. He noted stocks are little changed so far this year, with the S&P barely half a percent higher.
"We're waiting for what we think will be a good jobs number," he said. "Economies are improving and ought to support a higher stock market."
Data was mixed on Tuesday. The major economic report expected for the week is the nonfarm payrolls report on Thursday, which could shed light on the timing of a rate hike. Markets are closed Friday for the July 4 holiday.
"I think there's talk that Greece will get some sort of deal done, although I don't think that's the case," said Robert Pavlik, chief market strategist at Boston Private Wealth.
Eurogroup President Jeroen Dijsselbloem said in a Reuters report that it is too late for an extension of the Tuesday deadline and that the institutions will only consider the request for a new program after the referendum on July 5.
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He added that the Eurogroup of finance ministers will meet again on Wednesday to discuss the new Greek proposal after holding a conference call earlier in the day.
The Greek government indicated in the call it could change its stance on the referendum it has called for Sunday if a request for a new loan could be agreed, euro zone sources told Reuters.
Separately, Finnish Finance Minister Alexander Stubb said the Eurogroup concluded that requests from Prime Minister Alexis Tsipras for an extension of Greece's bailout program or debt relief were not possible, Reuters reported.
The prime minister said the government will seek a viable solution until the end and aims to stay in the euro, Reuters reported. The Greek government also issued a statement saying it has prepared a two-year agreement with the euro zone bailout fund in order to cover its financing needs, Dow Jones reported.
German Chancellor Angela Merkel said on Tuesday that Germany would not negotiate on a new bailout agreement for Greece before its referendum which is planned for Sunday.
"There's still a downward trend in place. At this juncture markets are holding relatively steady, otherwise holding their breath until the weekend," said Eric Lascelles, chief economist at RBC Global Asset Management. "The market's default assumption is that Greece does miss that deadline."
The Dow and S&P both briefly turned negative in midday trade amid reports that S&P downgraded four of Greece's banks to "selective default." The indices turned higher after struggling to hold slight gains.
"The Greece situation continues to grind on. It's not encouraging. (But) you had a selloff and a pause. It suggests not panicking," said Bruce McCain, chief investment strategist at Key Private Bank.
"It's more the assessment that this is not the crisis with Lehman Brothers or other situations that lead to broader systemic problems," he said.
Analysts also noted some positioning as Tuesday marks the end of the second quarter and first half of the year.
"I think it's an oversold bounce. ... We also had a lot of nervousness (yesterday)," said Adam Sarhan, CEO of Sarhan Capital. "But from a market standpoint you really want to see the bulls step up and buy this dip. If they don't, then when will they?"
Stocks had their worst day for the year so far on Monday, with the Nasdaq plunging 2.4 percent, the Dow Jones industrial average closing below its 200-day moving average and the S&P 500 barely above its 200-day moving average.
"Next support is 2044-2054 based on both the 200-day moving average and the March low," BTIG Chief Technical Strategist Katie Stockton said in a note. "Our indicators were mostly constructive ahead of the pullback, but the negative reaction to the news out of Greece certainly heightens risk in the near term for the SPX."
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"Today's trading is showing no signs of a sustainable bounce from yesterday's tarnish," Peter Cardillo, chief market economist at Rockwell Global Capital, said in a Tuesday note. "This action suggests a test to the lower support areas is in sight."
European stocks extended Monday's selloff to end more than 1 percent lower as Greece was expected to default later that night.
The Greece stock exchange and banks remained closed on Tuesday.
"This temporary noise from Greece will blow over and investors can focus on (good) fundamentals," said Doug Cote, chief market strategist at Voya Investment Management. "The bond market is indicating that there's no real concern about Greece."
He expects Greece will reach an 11th-hour deal with its creditors.