The latest economic data doesn't point to a major resurgence in payrolls coming later this week, potentially complicating the Federal Reserve's rate decision later this month.

Fed Chairman Kevin Warsh gave a hawkish speech at the annual Jackson Hole Economic Policy Symposium on Friday. That recalibrated the markets' expectations in favor of a rate hike at the September policy meeting after a recent bout of cooler inflation data had dampened expectations that the Fed would raise interest rates.

"And while this summer's [inflation] readings were better than expected, they do not tell me that underlying trends have meaningfully improved," Warsh said, noting that the Federal Open Market Committee stands ready to act as circumstances might require.

But while inflation will likely remain the driver behind the September rate decision, investors shouldn't dismiss the possibility that the latest raft of employment data could shift expectations again. Traders are now pricing in 68% odds of a rate hike as of Tuesday, according to the CME FedWatch tool. Yet Friday's jobs report could revive fears that labor conditions are not strong enough to handle higher interest rates.

"While inflation data will be most important this month, another soft jobs report could be especially important in the decision making for Fed officials who are less concerned than others that the economy is reheating and leading to sticky inflation," writes Citi economist Veronica Clark.

The Bureau of Labor Statistics is set to release its latest employment data on Friday. Economists surveyed by FactSet expect employers to have added 65,000 jobs in August after payrolls fell by 23,000 in the initial July estimate. The national unemployment rate is projected to rise to 4.2% from July's 4.1%.

But there's a wide disparity among the payroll estimates, with forecasts ranging from a decline of 25,000 from Fifth Third Bank's Bill Adams to a gain of 121,000 from Regions Bank. Several major financial firms, including Bank of America, Barclays, Citi, Goldman Sachs, Nationwide, RBC Capital Markets, and Vanguard, expect payroll gains of 40,000 or less.

The key question, however, is whether the latest jobs data will contradict Warsh's assessment that the U.S. labor market is doing well. "Labor markets are quite stable," Warsh said Friday, signaling that employment conditions could withstand higher interest rates if need be.

August payroll gains have come in softer than expected over the past two years, largely due to residual seasonality in the data. ADP's labor data have also been soft, showing a steady decline in the four-week moving average since the end of June.

In early August, however, weekly payroll data did jump higher, though it remains to be seen whether that trend continued throughout the month. The ADP National Employment Report for August will be released Wednesday morning.

"Similar to July, summer seasonality may have weighed on other sectors in August as well, likely limiting the expected rebound from July's outsized declines in local government education jobs and leisure & hospitality," writes Shruti Mishra, U.S. economist for Bank of America Securities. She estimates that payrolls will advance by just 40,000 in the August jobs report.

Vanguard, meanwhile, estimates that private-sector payroll growth slowed to just 8,000 jobs in August based on data that tracks new enrollments and separations in Vanguard-administered 401(k) retirement plans. Net new hires were down in August in Vanguard's data, especially among companies with fewer than 250 employees. Payroll processor Gusto also found that small-business hiring slipped below the 12-month average in August, adding just 29,500 net new jobs.

The expiration of temporary protected status (TPS) work authorization for roughly 200,000 Haitian nationals could also affect August payrolls and unemployment, writes EY-Parthenon chief economist Gregory Daco. The number of workers directly affected by the status change remains uncertain, but it could create a temporary drag on employment, particularly in the services sector.

Tuesday's Job Openings and Labor Turnover Survey also showed little evidence of a hiring resurgence. The hires rate, which had shown modest signs of a revival in recent months, dropped by 0.2 percentage points to the lowest level since February.

There were 7.3 million jobs available at the end of July, a bit below the 7.31 million positions expected, but up slightly from June's revised level of 7.18 million, according to data released Tuesday by the BLS. The job-opening rate remained below the recent highs reached in April and May, pointing to some "genuine renewed softness" in the labor market, writes Capital Economics economist Stephen Brown.

The quits rate remained low, suggesting that workers are still unwilling to risk changing jobs-especially in sectors such as construction and information technology. Layoffs, however, remain low, while initial jobless claims have also been benign.

The Institute for Supply Management's manufacturing purchasing managers index, or PMI, also failed to provide much comfort on Tuesday. The latest reading came in at 54.6 for August, a pullback from 55.6 in July that was driven in part by a sharp fall in the new orders component.

Employment in the manufacturing sector, which had ticked up last month, fell by 1.6 percentage points to 51.2 in August. But the component measuring prices paid by manufacturers held firm at 71.1. That indicates input costs, such as energy, commodities, and semiconductors, continue to increase at a rapid pace and makes the case for near-term rate hikes more compelling, says James Knightley, chief international economist at ING.

Given the pricing pressures still in the pipeline, the August jobs report may have to come in considerably weaker than expected to materially shift rate expectations following Warsh's speech.

But investors should be aware that the risk of such a downside surprise isn't insignificant.