Reports stated that the letter pledges that the US will take back the weapons supplied once the joint operation is over.
Ara News quoted military sources as saying that the Kurdish People's Protection Units (YPG) and allied factions of the Syrian Democratic Forces (SDF) have brought military reinforcements to the northern countryside of Aleppo province to impede the Turkish progress towards Afrin district.
Both the YPG and the Turkish government claim to be fighting the Islamic State in Syria.
The Kurdish People's Protection Units, or YPG, are a key part of the USA -backed Syrian Democratic Forces.
However, the US views the PYD/PKK as its ally in the fight against Daesh in northern Syria. While the United States supports the YPG and sees it as a key partner in the fight against IS, Turkey considers it a terrorist organization linked to Turkish-Kurdish separatists inside Turkey, known as the PKK.
The Turkish military said that it had responded to fire from the YPG, the Kurdish militia that controls the area.
Turkish President President Recep Tayyip Erdogan has warned in recent days that Turkish troops could cross into northern Syria to prevent the YPG from helping Kurds to form what he called "a state".
"This is a message from the United States to Turkey to stop attacking the YPG, since they fear this could threaten the anti-ISIS Raqqa operation", local Kurdish journalist Heybar Othman told ARA News.
Mattis' comments marked the first time he has spoken at length publicly about the us pledge to recover the weapons provided to the Kurds. "In the first two weeks [of the campaign], there was significant progress-very quick progress that was made... the SDF in the advance have since hit some significant resistance from ISIS", U.S. coalition spokesman Col. Ryan Dillon said on Sunday.
In a letter sent to Turkish counterpart, Fikri Işık last week, US Defense Secretary James Mattis stated that the US will regularly share the list of materials that YPG supplies with Turkey every month.