U.S. Treasury prices edged higher, supported by a slight decline in crude oil prices during the American trading session, leading to a bull-flattening trend in the yield curve. Shorter-dated Treasuries found early support following the release of the ISM Services PMI's prices paid index, which came in below expectations. Although swap contracts moved away from the most hawkish levels reached on Monday, market pricing still reflects expectations that the Federal Reserve could potentially raise interest rates by mid-next year.

Shortly after 3 p.m. in New York, yields on the long end of the curve fell by as much as 3 basis points, with the 30-year yield retreating below the 5% threshold. Gains in long-term bonds drove a narrowing of the 2s10s and 5s30s spreads by approximately 0.5 basis points and nearly 2 basis points, respectively, over the session.

Overnight indexed swaps tied to Fed meeting dates indicate that markets are pricing in roughly 15 basis points of tightening by the April meeting next year, down from around 17 basis points at Monday's close. As oil prices declined, swap markets trimmed a small portion of their rate-hike expectations.

In late trading, WTI futures fell nearly 4% as a fragile ceasefire in the Middle East held and the United States downplayed the prospect of a full-scale conflict with Iran. Breakeven inflation rates came under pressure alongside lower oil prices, with the 5-year measure falling from around 2.75% to 2.68%.

Earlier in the day, U.S. Treasuries also reacted to a series of economic data releases, including the ISM services prices paid index, which registered 70.7, well below the forecast of 73.5.

As of 4:11 p.m. Eastern Time, the 2-year Treasury yield stood at 3.9357%. The 5-year yield was at 4.0759%, while the 10-year yield reached 4.4231%. The 30-year yield settled at 4.9907%. The yield spread between 2-year and 10-year notes was 48.54 basis points, and the gap between 5-year and 30-year yields measured 91.3 basis points.