WASHINGTON—Senate Majority Leader Mitch McConnell urged President Trump to “exercise American leadership” and keep U.S. troops in northern Syria, joining a chorus of Republican and Democratic lawmakers who warned the president against withdrawing from the area.

“American interests are best served by American leadership, not by retreat or withdrawal,” the Kentucky Republican said. He didn’t directly criticize Mr. Trump, but reminded him that the Senate voted in January to keep U.S. troops in Syria, in reaction to a previous withdrawal plan announced by the White House.

The remarks by Mr. McConnell came as many Republicans broke publicly with Mr. Trump, including some who rarely speak out against the president. Many painted their disapproval in stark terms, and floated ways to punish Turkey if it moved ahead with a military incursion in the region.

Sen. Lindsey Graham (R., S.C.), a close ally of Mr. Trump, called the pullout “a disaster in the making.” He warned it would ensure a comeback by Islamic State, or ISIS, and “will be a stain on America’s honor for abandoning the Kurds.”

The blowback came as American military personnel began moving away from the Syria-Turkey border, clearing the path for Turkey to launch an offensive against U.S.-backed Kurdish fighters. The Kurds have led the fight against Islamic State militants alongside the U.S., but Ankara considers them terrorists.