U.S. Treasury yields turned lower in European trade while eurozone government bond yields fell sharply as investors pondered whether the recent selloff had gone too far and now offered attractive levels to start buying again.

Reduced market pricing of a Federal Reserve interest-rate hike later this month following Friday's weak employment data has helped calm nerves. Investors priced a 22% probability of a quarter-point Fed rate increase on Oct. 28, sharply down from as high as 70% at the start of last week, although at least three rate hikes are priced over the coming year, LSEG data showed.

French government bonds recovered after a sharp selloff due to recent budget concerns that took yields and their spreads against German peers to multiyear highs.

"Markets have had another volatile session over the last 24 hours, as investors grappled with European contagion risk and a fresh Treasury selloff," Deutsche Bank strategists said. "On the bright side, yesterday brought some initial signs that the pressure on France was stabilizing, with a clear outperformance in French debt."

The 10-year Treasury yield fell 3 basis points to 5.280%, having risen to 5.349% on Monday, the highest since 2002, according to Tradeweb. The 10-year German Bund yield declined 5.1 basis points to 3.445%. French and Italian bonds rallied, with the 10-year French OAT yield sliding 12 basis points to 4.758% and the Italian BTP yield declining 13 basis points to 4.543%.

The 10-year French OAT-Bund yield spread narrowed to 131 basis points, having widened to an intraday high of almost 159 basis points on Friday.

That said, government bond yields globally remain elevated.

Government spending, debt sustainability and inflation remain key concerns, largely due to steep rises in energy prices as a result of the U.S.-Iran war.

Regarding France, RBC Capital Markets analysts said that the passing of the 2027 budget doesn't necessarily secure fiscal sustainability.

"Beyond next year, France would need to deliver a similar pace of consolidation for four further years to achieve a 3% budget deficit by 2030--a pace no French government has sustained in the modern era," they said in a note.

This week sees less eurozone government bond supply, with no French auctions due.

The U.S. Treasury's $58 billion three-year auction on Tuesday is expected to be absorbed well.

"Given a more supportive broader macro and technical backdrop, we think [Tuesday's] auction will be digested smoothly," J.P. Morgan strategists wrote in a note.

In Europe, Austria and Germany will hold bond auctions on Tuesday, with the former reopening 2032- and 2062-dated bonds and the latter launching new December 2028-dated treasury notes, or Schatz.