* Private payrolls up 173,000 vs est. 175,000
* Weekly jobless claims unexpectedly fall
* ECB keeps interest rates unchanged
* Box falls after billings growth slows
* Futures down: Dow 19 pts, S&P 2.75 pts, Nasdaq 3.5 pts (Adds details, comment, updates prices)
By Yashaswini Swamynathan
June 2 (Reuters) - Wall Street was set to open slightly lower on Thursday, as data showing the U.S. labor market continued to strengthen even as the European Central Bank maintained negative interest rates.
The ADP National Employment report showed U.S. private payrolls increased a less-than-expected 173,000 in May. But, as the economy nears full employment, job creation would slow, said Mark Zandi, Moody's Analytics' chief economist.
The ADP data comes ahead of the U.S. Labor Department's more comprehensive non-farm payrolls report on Friday, which includes both public and private-sector employment.
The number of Americans applying for unemployment benefits unexpectedly dropped last week, pointing to a tightening labor market. The four-week moving average of claims, considered a better measure of labor market trends, also fell.
The ECB kept interest rates unchanged and President Mario Draghi said inflation would likely remain very low or negative in the next few months.
Most of the European markets were in the red on Thursday.
Contrasting data from the United States and abroad has left investors mulling if the Federal Reserve will, as was broadly expected earlier, raise rates as soon as this month.
Traders are now pricing in an 18.8 percent chance of a hike this month, significantly lower than their expectations on Monday, while chances of a hike in July have increased to 50 percent, according to CME Group's FedWatch tool.
"Obviously there are several major events that the market is going to focus on and could cause a bumpy ride for stocks today," said Peter Cardillo, chief market economist at First Standard Financial in New York.
Dow e-minis were down 19 points, or 0.11 percent at 8:27 a.m. ET (1227 GMT) with 17,674 contracts changing hands.
S&P 500 e-minis were down 2.75 points, or 0.13 percent, with 130,730 contracts traded.
Nasdaq 100 e-minis were down 3.5 points, or 0.08 percent, on volume of 14,430 contracts.
Wall Street eked out minor gains on Wednesday as stronger-than-expected U.S. factory data helped negate crude oil prices that oscillated due to concerns about whether or not the OPEC would cap output.
Oil prices were flat the OPEC meet to decide on output got underway.
Oracle was down 3.1 percent at $39 premarket after it was sued by a former senior finance manager who claimed she was terminated in retaliation for complaining about improper accounting practices.
Cloud storage provider Box slumped 12.2 percent to $11.25 after the company reported a slowdown in billings growth in the first quarter.
Apple was down 0.7 percent at $97.70 after Goldman Sachs cut its price target on the stock, citing lower growth in the smartphone industry. (Reporting by Yashaswini Swamynathan in Bengaluru; Editing by Savio D'Souza)