U.S. and European government bond yields mostly rose Tuesday, driven by a renewed increase in oil prices, lingering concerns over budget deficits and supply pressure from both government and corporate bonds.

Investors, meanwhile, anticipate diplomatic progress in the U.S.-Iran conflict after U.S. President Trump signaled his readiness to meet his Iranian counterpart Masoud Pezeshkian at the UN General Assembly meeting this week. Investors also look forward to a meeting between Trump and Chinese President Xi Jinping later in the week.

Budget deficit issues continue to exert an 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," Mohit Kumar, global economist at Jefferies 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.

The 10-year U.S. Treasury yield rose 0.7 basis point to 4.969%, according to Tradeweb. The 10-year German Bund yield increased 2 basis points to 3.469% and the 10-year French OAT yield rose 6.3 basis points to 4.522%, causing the 10-year OAT-Bund spread to widen to 105.1 basis points. The 10-year U.K. gilt yield was up 3.5 basis points at 5.239% following higher-than-expected public sector borrowing in August.

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.

Tuesday's yield increase comes ahead of the U.S. Treasury's $69 billion auction of two-year Treasurys.

"Overall, while we are sensitive to the high level of outright yields, we ultimately think the macro backdrop will weigh on the supply process, and the auction will likely require more of a concession in order to be digested smoothly," J.P. Morgan rates strategists 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.