Only one of the 10 voting members of the Federal Open Market Committee was ready to raise the benchmark short-term rate at the June 16-17 meeting. But that unnamed official "expressed a willingness to wait another meeting or two for additional data," the minutes said.
The so-called federal funds rate has been near zero percent since December 2008 in an attempt to boost economic growth.
Fed Chairwoman Janet L. Yellen told reporters after the June meeting that she expected a rate hike this year, and many analysts believe it will happen in September.
But trouble with the global economy could delay action until later in the year.
Fed policymakers were worried last month about problems in Europe and emerging market economies such as China -- and those situations have worsened.
Greek and European leaders are struggling to agree on terms of a new bailout for the nation while stock prices in China have plummeted in recent weeks.
"Many participants expressed concern that a failure of Greece and its official creditors to resolve their differences could result in disruptions in financial markets in the euro area, with possible spillover effects on the United States," the minutes said.
At the June meeting, Fed policymakers downgraded their view of the economy based on an economic contraction in the first quarter that was caused by temporary factors, "including unfavorable weather in parts of the country and labor disputes at West Coast ports," the minutes said.
The officials expected economic growth to get back on track in the second half of the year. But they were concerned that consumers were spending cautiously and low energy prices and the high value of the dollar were holding back some sectors of the economy.
Although some policymakers thought the economy was close to full employment, most said there was room for more improvement in the labor market, the minutes said.
Last week, the Labor Department reported strong growth of 223,000 net new jobs in June and the
The Fed is mandated to use monetary policy to try to maximize employment and keep prices stable. Inflation has been running well below the Fed's 2% annual target and all but one of the policymakers said they needed more evidence that the economy was growing fast enough to push prices toward that level.