By Andrew Gray, Lili Bayer, Phil Stewart and Idrees Ali

BRUSSELS (Reuters) - U.S. Defense Secretary Pete Hegseth rejected criticism on Thursday of Washington's negotiating strategy with Russia on the war in Ukraine and warned Europe against treating America like a "sucker" by making it responsible for its defense.

"Make no mistake, President Trump will not allow anyone to turn Uncle Sam into 'Uncle Sucker,'" Hegseth told a press conference at NATO headquarters in Brussels.

Europe should be primarily responsible for defense on the European continent, he said.

In his overseas debut after taking the helm charge of the Pentagon on Jan. 24, Hegseth has set off a storm of criticism in Europe after announcing on Wednesday that a return to Ukraine's pre-2014 borders was unrealistic and the Trump administration does not see NATO membership for Kyiv as part of a solution to the war triggered by Russia's 2022 invasion.

Hegseth's critics say the remarks amounted to the United States giving up its leverage in negotiations with Russia before they even started, in what they described as a major victory for Russian President Vladimir Putin.

Hegseth on Thursday defended his remarks as acknowledgement of battlefield realities and left the door open for U.S. moves during negotiations that could be seen as concessions, including on issues like U.S. supplies to Ukraine and troop levels in Europe.

"I think it would be fair to say that things like future funding, either less or more, could be on the table in negotiations," he told reporters, adding he would not get ahead of U.S. President Donald Trump.

Trump spoke with Russian President Vladimir Putin and Ukrainian President Volodymyr Zelenskiy on Wednesday and told U.S. officials to begin talks on ending the nearly three-year-long war.

Trump's unilateral overture to Putin, accompanied by apparent concessions on Ukraine's principal demands, raised alarm for both Kyiv and the European allies in NATO who said they feared the White House might make a deal without them.

"We, as a sovereign country, simply will not be able to accept any agreements without us," Ukrainian President Volodymyr Zelenskiy told reporters in Ukraine.

European officials took a firm line in public towards Trump's peace overture, saying any agreement would be impossible to implement unless they and the Ukrainians were included in negotiating it.

"Any quick fix is a dirty deal," European foreign policy chief Kaja Kallas told reporters at NATO headquarters earlier in the day.

Hegseth rejected the idea that he was undermining NATO or European security.

"NATO is a great alliance, the most successful defense alliance in history, but to endure for the future our partners must do far more for Europe's defense. We must make NATO great again," he said.

(Reporting by Andrew Gray, Lili Bayer in Brussels, Phil Stewart and Idrees Ali in Washington; Editing by Chizu Nomiyama and Howard Goller)