A push towards Manbij, in a separate Kurdish-held enclave some 100 km (60 miles) east of Afrin, could threaten US plans to stabilize a swath of northeast Syria.
A key USA negotiating position is that there must be a political transition away from Syrian President Bashar al-Assad, whom Washington blames for the brutal conflict and accuses of using chemical weapons on his people.
US Defence Secretary Jim Mattis urged Turkey on Tuesday to exercise restraint in its military operations in northern Syria, which he said had disrupted the peaceful return of refugees and could prove to be an opening for al Qaeda and Islamic State.
Two civilians died in a hospital from the injuries after a missile fired from Afrin hit the mosque in the center of Turkey's Kilis located on the border with Syria, the DHA agency reported.
Erdogan seemed critical of Trump's comments and asked the United States to halt weapons support to the Kurdish YPG militias in Syria, Erdogan's office said.
Syrian Kurdish forces and their allies have set up three autonomous cantons in the north, including Afrin in the northwest, since the start of the Syrian conflict.
Erdogan said Turkish troops and allied fighters have killed at least 268 Syrian Kurdish fighters since the operation began, a number disputed by Kurdish officials and a war monitoring group.
In an interview with Reuters, Turkey's government spokesman said he saw a small possibility that Turkish forces could come face-to-face with the USA troops in Manbij.
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Turkey says it wants to create a 30 kilometer (20-mile) deep "secure zone" in Afrin.
The rocket attack came after Turkey launched an operation in Syria's Afrin against the PKK/KCK/PYD-YPG and Daesh terrorist groups on Saturday.
The Turkish military said it carried out the operation under the framework of Turkey's rights based on global law and it was respectful of Syria's territorial integrity.
A senior US official said that as of Tuesday the Turks had not been ready to engage in detail on such a proposal.
"Turkey has not been able yet to shore up its control over any of the villages it has advanced on", said the Observatory's Abdulrahman. He said the United States helped Kurdish fighters to liberate Manbij and has promised to keep defending it.
Syrian Kurdish forces captured Manbij, which is west of the Euphrates River, from IS in 2016 with the help of the USA -led coalition.
"It's a headache", said Macer Gifford, a British man who formerly volunteered with the YPG in the fight against Isis but who is now home in the UK.
The United States has hoped to use the YPG's control of territory to give it the diplomatic muscle it needs to revive UN-led talks in Geneva on a deal that would end Syria's civil war and eventually lead to Assad's removal.