U.S. Treasury and European government bond yields rose Tuesday, while President Trump threatened to escalate the conflict with Iran if Tehran doesn't agree to a deal.

Trump said at the U.N. General Assembly that he faces the dilemma of striking a deal with the Islamic republic or to "annihilate" it. "Do I drive them into hell with no chance of survival and no hope of future greatness or generations?" Trump asked.

The speech came as oil prices fell on hopes of a diplomatic solution for the Strait of Hormuz.

On top of that, a $69 billion auction of two-year Treasury notes cleared at the highest yield since May 2024, amid robust demand indicators. The Treasury will auction $70 billion in five-year notes Wednesday and $44 billion in seven-year notes Thursday.

Treasury yields rebounded from early declines and traded near Monday's levels, with the two-year at 4.751% and the 10-year at 4.963%.

Yields drifted amid a lack of major economic indicators, tracking oil prices for most of the session, according to Eric Winograd, chief economist at AllianceBernstein.

"On a day like today when there is no other driver . . . yields tend to drift around [and] there's a correlation with oil prices," he said.

Winograd expects yields to remain near current levels, amid lingering concerns about U.S. government spending and elevated energy costs.

The 10-year German Bund rose slightly to 3.447% from 3.434%. The 10-year French OAT yield moved up to 4.490% from 4.477%. The 10-year U.K. gilt yield increased to 5.226% from 5.179%.

Budget-deficit issues continue to exert upward pressure on bond yields, or at least prevent them from significant falls.

"Even if oil prices move lower, we do not think that long end rates could rally significantly," Jefferies global economist Mohit Kumar said in a note. "The deficit picture, particularly in Europe, is concerning as we head into the budget discussions in Q4 and elections next year," he said.

However, the 10-year French OAT-German Bund yield spread is expected to consolidate around 100 basis points as the market adjusts to this new reality, Chris Attfield, European rates strategist at HSBC, said in a note.

"There is little point in trying to draw 'lines in the sand' beyond which spreads will not rise, but 120 bps may be the next psychological level in any further spread weakness," he said. Eurozone yield spreads have been systematically widening in response to the surge in yields, but France's move is far larger than HSBC would expect given its debt-to-GDP ratio, he said.

Supply in Europe comes from Germany auctioning 5 billion euros ($5.73 billion) in October 2031-dated federal notes, or Bobl, and Italy syndicating a new October 2038-dated green BTP.