July 8 (Reuters) - The euro slipped on Monday, but came off of overnight lows against the dollar after France's election pointed to a hung parliament.

The U.S. dollar crept up from a more than three-week low after U.S. payrolls data on Friday boosted bets that the Federal Reserve will soon start cutting interest rates.

French President Emmanuel Macron on Monday asked his prime minister to stay in the role for now, pending what will be difficult negotiations to form a new government after a surprise left-wing surge in elections that delivered a hung parliament.

“We're still waiting to see if the coalition can get the 240 to 250 lawmakers together to have any semblance of a, what is it, I think in France, a working government. We're in wait-and-see mode there,” said Garth Appelt, Head of Foreign Exchange & Emerging Markets Derivatives, Mizuho Americas in New York.

Some concerns about a potential French exit from the eurozone were also eased after eurosceptic Marine Le Pen’s National Rally failed to win a majority, said Helen Given, FX trader at Monex USA in Washington DC.

"There was a small risk that France would actually start to move towards exiting the Eurozone" if the National Rally had won, Given said. "People are just happy to have it off the table."

The euro was last down 0.11% at $1.0824. It briefly reached $1.0845, the highest since June 12, after dipping to a low of $1.07915 earlier in the day.

The dollar index, which measures the U.S. currency against the euro, sterling, yen and three other major rivals, gained 0.04% to 104.99. It earlier fell to 104.80, the lowest since June 13.

The greenback fell on Friday after June's employment report showed solid jobs gains in the month, but softer details under the hood.

Government and healthcare services hiring made up about three-quarters of the payrolls gain and the unemployment rate hit a 2-1/2-year high of 4.1%.

The economy also created 111,000 fewer jobs in April and May than previously estimated, while annual wages increased at the slowest pace in three years.

Traders will pay close attention to comments by Fed Chairman Jerome Powell when he testifies before Congress on Tuesday and Wednesday for signs that a rate cut is getting closer.

Traders see two cuts this year as likely and are pricing in a 76% likelihood of this first rate cut at the Fed's September meeting, with a subsequent cut expected by December, according to the CME Group's FedWatch Tool.

This week's main U.S. economic release will be consumer price data for June on Thursday.

The dollar gained slightly against the Japanese yen, following data earlier on Monday showing that Japanese workers saw their average base pay climb 2.5% in May, the fastest pace in 31 years.

The Bank of Japan said wage hikes were broadening across the economy due to tight labor market conditions, signaling its confidence the country was making progress toward durably achieving its 2% inflation target.

The optimistic assessment may heighten the case for the central bank to raise interest rates as soon as its next meeting on July 30 to 31.

“You're starting to see more and more discussions about the BOJ lifting off a little bit in terms of hiking policy. They're thinking that the BOJ is getting much closer to the hiking window, whether it be this month or the next meeting," said Appelt.

The dollar was last up 0.02% at 160.76 yen, after last week reaching a 38-year high of 161.96.

Sterling was steady, after earlier reaching a 3-1/2 week top versus the dollar and the euro. The British currency has gained since the Labour Party's landslide election victory last week, which ended 14 years of Conservative rule. It was last up 0.02% at $1.281.

The Aussie dollar fell 0.18% versus the greenback at $0.6737, having earlier reached $0.67615, the highest since Jan. 3.

In cryptocurrencies, bitcoin fell 0.08% to $56,312.

(Reporting By Karen Brettell; Additional reporting by Kevin Buckland and Amanda Cooper; Editing by Arun Koyyur and Josie Kao)