A U.S. State Department official said such moves would undermine regional stability and would not help protect Turkey's border security.
The operation is thought to be targeting Kurdish organizations Kurdistan Workers' Party (PKK) and its affiliates Democratic Union Party (PYD) and People's Protection Units (YPG).
The US-backed YPG says at least nine people were killed in the air strikes.
Turkey's military says it has begun attacking Kurdish-held positions in Syria's Afrin region, including reportedly firing on positions held by a group that the US has backed in the past.
The Russian Foreign Ministry on January 20 voiced concern about the new offensive and called on "the opposing parties to show restraint". The U.S. recognizes the PPK as a terrorist outfit, but the militias in Afrin are associated with the YPG - an entirely different organization that has fought side by side with the U.S.in battling both ISIS and insurgents in Iraq.
The Turkish defense minister described the artillery attacks as the "de facto start" of a publicly touted planned invasion of Afrin.
Cavusoglu said Turkey wasn't satisfied with the US explanation, while adding that it has no interest in colliding with the USA over the operation.
The United States and European is backing the YPG although it considers the PKK as a terrorist group.
Military vehicles and work machines sent to reinforce the troops stationed at the Syrian border were already in Turkey's Hatay province.
The Turkish Foreign Minister stated that Turkey had notified Russia, Syria, the United Nations and the USA about the operation.
But the YPG has been the key ally of Turkey's fellow North Atlantic Treaty Organisation member the United States in the fight against IS jihadists, playing a key role in pushing the extremists out of their Syrian strongholds.
The FSA spokesman did say the rebel group backs the Trump administration's Syria policy unveiled by Secretary of State Rex Tillerson this week, saying it is a "good start".