(Adds missing word "the" in first paragraph, corrects nickname for the Swedish krona in paragraph 5, corrects verb tense in paragraph 6)

By Chibuike Oguh

NEW YORK (Reuters) -The dollar hit a near four-year low against the euro on Monday amid worries over the rising U.S. government deficit and uncertainty surrounding trade deals with major countries.

Senate Republicans will try to pass President Donald Trump's sweeping tax-cut and spending bill, despite divisions within the party about its expected $3.3 trillion hit to the nation's debt pile.

The dollar dropped 0.63% to 0.79355 against the Swiss franc, on track to end the month down 3.60%. The greenback has lost about 12.5% against the Swissie this year.

The euro hit its highest against the dollar since September 2021 at $1.1780. It was last up 0.45% and set to gain about 3.8% for the month. The single currency has gained about 14% against the dollar this year.

"There's a lot of focus around the big, huge bill and whether that gets approved," said Amo Sahota, executive director at FX consulting firm Klarity FX in San Francisco. "The dollar has been on a weakening trend. We are halfway through the year and the big winners have been the stocky (Swedish krona), the Swiss franc, and the euro. The euro's fortunes turned after the euro zone announced a huge spending bill."

The EU is open to accepting a trade agreement with the US that would apply a universal 10% tariff on many of its exports, Bloomberg News reported on Monday.

Treasury Secretary Scott Bessent said that countries could still face sharply higher tariffs on July 9 even if they are negotiating in good faith, adding that any potential extensions will be up to Trump.

The U.S. and China had resolved issues around shipments of Chinese rare earth minerals and magnets to the United States, further modifying a May deal in Geneva, Bessent had said last week.

"You have a weak dollar due to a potentially large increase in our budget deficit, and you have continued uncertainty around these tariff deals," said Eugene Epstein, head of structuring for North America at Moneycorp in New Jersey. 

"We had this positive news from the EU for a little bit and we had potential positive deals coming up, but then you had Trump doing a temporary about-face on Friday on Canada and so forth," Epstein said.

Trump said Japan would be among countries to receive a trade letter outlining tariffs they would need to pay to the U.S.

The dollar was down 0.36% to 144.45 against the Japanese yen, on track to finish the month flat versus the Asian currency.

Canada halted its plans to begin collecting a new digital services tax targeting U.S. technology firms just hours before it was due to start on Monday in a bid to advance stalled trade negotiations with Washington.

The Canadian dollar strengthened against the U.S. currency on the session. It was set to notch its fifth straight month of gains against the greenback. The loonie was up 0.41% versus the greenback to C$1.353 per dollar.

The dollar index, which measures the greenback against a basket of currencies including the yen and the euro, fell 0.35% to 96.86, on track for its sixth straight month of losses. It is set to mark its worst half-year since the 1970s.

"It's kind of rotating a game of musical chairs, whether it's the 'big beautiful bill', the trade deals, and then the Iran-Israel conflict. It's all like taking turns to be at center stage; once one thing passes and the other thing is focused on," Epstein said.

The Swedish krona strengthened 0.48% versus the dollar to 9.462. Sterling strengthened 0.04% to $1.3719. It is up 2% in June.     

(Reporting by Chibuike Oguh in New York. Additional reporting by Kevin Buckland and Johann M Cherian. Editing by Mark Potter and Marguerita Choy)