BRUSSELS—European Union authorities won’t formally seek more deficit trimming from France and Italy in preliminary reviews of eurozone budgets this week, European officials say, though they are expected to warn that the two budget plans risk going off track.

The anticipated decisions are a sign the bloc isn’t adopting a hard line against Paris and Rome amid the region’s continuing economic woes. With unemployment in double digits and the eurozone at risk of slipping into another recession, authorities here are wary of requiring the eurozone’s second and third largest nations to hit previously agreed budget targets for 2015 through significant tax increases and spending cuts.

The reviews, which will likely be announced on Thursday, are part of the eurozone’s new system of budget oversight. Officials in Brussels at the European Commission, the EU’s executive arm, get to review eurozone budgets before they are adopted by national parliaments. If they identify major problems, authorities can issue formal recommendations that can lead to fines against governments that refuse to follow them.

Because of the eurozone’s flagging growth prospects, France and Italy have both proposed to miss budget targets they had agreed to earlier this year. Weaker-than-expected growth lowers tax revenue and boosts spending on social benefits such as unemployment insurance, making government budget-deficit targets harder to achieve.

Late last month, EU officials decided against raising serious objections to the French and Italian budget plans. On Thursday, the EU won’t make further recommendations for Paris and Rome, officials familiar with the decisions said. That partly reflects the fact that their budget plans for next year aren’t completed yet, nor are final budget results available for this year.