U.S. Treasury yields continued to fall on Thursday as investors focused on the latest developments in the Middle East, and their potential impact on the outlook for inflation and interest rates.

The 10-year U.S. Treasury note yield — the key benchmark for U.S. government borrowing — was more than 2 basis points lower in the early hours, at 4.3280%.

The 2-year Treasury note yield, which tends to react in line with short-term Federal Reserve interest rate decisions, fell by more than 2 basis points to 3.8469%. Meanwhile, the longer-dated 30-year Treasury bond yield was 4.9204%, a 2 basis point drop.

One basis point is equal to 0.01%, and yields and prices move in opposite directions.

The fall in the cost of borrowing comes amid reports that the U.S. and Iran may be close to agreeing an end to the two-month-long conflict.

President Donald Trump said Wednesday that Iran would be bombed "at a much higher level" if Tehran doesn't agree to a peace deal. Oil prices traded lower on Thursday, with West Texas Intermediate futures sliding 2.4% to $92.85.

Traders continue to assess how the duration of the conflict is weighing on inflation expectations, growth forecasts and the outlook for Fed rate decisions.

Ahead of April's nonfarm payrolls and unemployment data, due later this week, the Department of Labor will publish weekly initial jobless claims numbers on Thursday. Quarterly nonfarm productivity and labor data is also expected from the Bureau of Labor Statistics.

Initial jobless claims fell by 26,000 for the previous week ending April 26, to 189,000, well below market expectations of 215,000.

On Wednesday, ADP data showed that private sector jobs grew by 109,000 during April, up from 61,000 new positions in March, and outweighing the 84,000 forecast by a Dow Jones poll of economists.

Laura Cooper, global investment strategist and head of macro credit at Nuveen, said the recent hawkish sentiment within the Fed is expected to persist over the coming months.

"This is still a backdrop where growth risks are going to increasingly materialize," Cooper told CNBC's "Squawk Box Europe" on Thursday. "In the near-term, the uncertainty around inflation will keep yields elevated."

Cooper said the dissent seen in the Federal Open Market Committee last week "draws a line in the sand" and points to further challenges in building a consensus on rate cuts ahead of the new regime at the Fed.