Two-year Treasury yields were set for their biggest monthly fall since March 2020 on Tuesday and benchmark 10-year yields for their first monthly drop since November, in another sign that investor focus is turning from high inflation to growth fears.

Those fears, softening economic data and indications that U.S. inflation may be peaking all helped push yields lower in May.

The bond rally has also been supported by traders ramping down their bets on the "terminal rate" -- where interest rates will peak this cycle -- to around 3% from over 3.3% in early May..

Two-year yields, which are sensitive to interest rates expectations, dropped 17 basis points in May, the first monthly fall since July 2021 and the largest since March 2020, the height of the COVID-19 pandemic turmoil in financial markets.

Benchmark ten-year Treasury yields were set to end the month 10 basis points lower, in their first monthly fall since November.

"That's the market being a bit wary of the outlook for growth and also mindful that how far the Fed is going to go is uncertain," said Chris Scicluna, head of economic research at Daiwa Capital Markets.

"Yields are down but I guess that's a correction for the exuberance we had a month ago. But fundamentally we've still got further to go (higher)," Scicluna added, expecting bond yields and breakevens -- market-based gauges of inflation expectations -- to rise again.

On Tuesday, Treasury yields rose, driven in part by hawkish comments from Fed Governor Christopher Waller on Monday.

Waller said he is advocating to keep 50-basis-point rate hikes on the table until substantial reductions are seen in inflation, winding back expectations that the Fed might pause for breath after hikes in June and July.

By 1223 GMT, 10-year yields were up 10 basis points (bps) to 2.855%.

Two-year yields were 8 bps higher at 2.57%, easing from an earlier one-week high of 2.589%.

After a public holiday in the U.S. on Monday, Treasuries were also catching up with a sharp rise in German bond yields after German consumer prices were reported to have increased at their fastest pace in half a century. Euro zone inflation rising to another record high in May on Tuesday pushed yields even higher.

Also adding upward pressure on yields was Brent crude futures advance above $123 a barrel after the European Union vowed to slash imports of Russian oil by year's end.

"The market seems to go through occasional gyrations at the moment," said Rob Carnell, head of research for Asia-Pacific at ING.

"We have been quite pessimistic, with yields down where they are, and equities having sold off, and now we are having a couple of days when we decide whether or not this is a bottom."

(Additional reporting by Alun John, Vidya Ranganathan; Editing Kirsten Donovan and Bernadette Baum)