The SDF, which comprises Kurdish and Arab militias, is supported by elite United States forces and air strikes from a US-led coalition.
Later, Mr. Cevik attempted to walk back his comments on social media regarding USA forces working with Kurdish militias.
The Obama administration had approved military support for the YPG and other Kurdish militias in early 2015, as they fought to drive the Islamic State from the Syrian town of Kobane, which straddles the Turkish border.
The issue is guaranteed to be front and center during a scheduled visit by Turkish President Recep Tayyip Erdogan to Washington next week.
Turkey views the Kurdish People's Protection Units, or YPG, as a threat and has criticized the United States for partnering with them in Syria. The official has no confirmed or disclosed just how soon arms would start being delievered to Kurdish fighters.
The YPG clashed with the Turkish military past year as Turkey began Operation Euphrates Shield to clear border areas of ISIS.
While White did not mention the kinds of arms to be provided to the Kurds, other officials had indicated in recent days that 120mm mortars, machines guns, ammunition and light armored vehicles were possibilities.
"We're keenly aware of the security concerns of our coalition partners in Turkey", said White House press secretary Sean Spicer on Tuesday. "I don't believe the Trump administration will do the same", Erdogan said. "We want to reassure the people and government of Turkey that the U.S.is committed to preventing additional security risks and protecting our North Atlantic Treaty Organisation ally".
Mattis would not elaborate on the possible Turkish involvement or if that meant Turkish troops would enter Raqqa.
"Our intent is to work with the Turks, alongside one another, to take Raqqa down, and we're going to sort it out and we'll figure out how we're going to do it", Mattis told a news conference, after defence talks with members of the USA -led coalition battling IS. "We're going to sort it out, we'll figure out how to do it, but we're all committed to it". "NATO allies stick together, " he said.
The U.S. officials who disclosed the Trump administration decision weren't authorized to publicly discuss the matter and demanded anonymity.
Turkey carried out a series of airstrikes against the YPG last month, a move that drew immediate complaints from USA officials. In the latest airstrikes, on Tuesday, Turkey said that it had destroyed "PKK terrorist camps" in northern Iraq, according to Turkey's semi-official Anadolu News Agency. The SDF is expected to spearhead the coming assault on the Islamic State capital of Raqqa.
Turkey allows the USA -led coalition fighting ISIS to use one of its air bases for warplanes involved in the fight, but one risk for the Americans is that Erdogan could withdraw that permission over his objections about the Kurds.
As many as 1,000 United States troops are working alongside the SDF inside Syria.