Eurozone leaders are due to meet at a summit later on Tuesday to discuss the issue, although reports have said the Greeks will not put forward new proposals until Wednesday.
Stocks had fallen on Monday after Greece rejected the terms of the latest bailout offer in a referendum.
In early trade on Tuesday, the Dow Jones fell 39.82 points to 17,643.76.
The S&P 500 index dropped 2.88 points to 2,065.88, while the Nasdaq slid 11.59 points to 4,980.35.
Shares in Advanced Micro Devices (AMD) dived nearly 18% after it said second-quarter revenues would fall by 8% from the previous quarter, a worse performance than previously estimated.
AMD blamed weak demand for PCs for the drop.