Dow futures were down about 180 points, off earlier lows of more than 220 points lower.
Greece's main stock exchange was closed on Monday and banks across the country were shut to prevent a run on financial institutions. The central bank has also recommended a 60 euro ($66) limit on withdrawals from cash machines.
The closures came after reforms-for-financial aid talks between Greece and its creditors broke down and the European Central Bank capped emergency funding to Greek banks on Sunday.
Greek Prime Minister Alexis Tsipras has also called a referendum on July 5, in which Greeks will vote on whether to accept the rescue package previous offered by Athens' international bailout supervisors, which is contingent on austerity measures.
"In the absence of a sudden dramatic turn of events towards reconciliation between the Greek government and its creditors, it is difficult to see how it will be able to repay the European Central Bank on July 20, whatever the outcome of the coming weekend's referendum," said Chris Scicluna, head of economic research at Daiwa Capital Markets Europe, in a research note on Monday.
The developments in Greece wreaked havoc on global markets, with stock markets in France and Germany falling over 4 percent. Japan's Nikkei closed down 2.9 percent. The Shanghai Composite also closed 3.3 percent lower despite initial attempts at gains following the central bank's rate cut over the weekend.
In the U.S., pending home sales data for May are due out late on Monday, but eyes will be fixed on the upcoming nonfarm payrolls employment report. These will come out a day early for Independence Day.
"Away from Greece, Thursday's U.S. nonfarm Payrolls data is the only data release of note which may influence market sentiment," said Jassie Singh, credit strategist at Lloyds Bank, in a research note on Monday.