The detention of a member of staff in Istanbul last week has triggered a diplomatic row, prompting both sides to halt most visa services.
The first worker was held over alleged links to a cleric blamed for the failed 2016 Turkish coup, but no reason has been given for the latest warrant.
The row has driven down Turkey's currency and stocks.
Turkey has asked Washington to review its decision to suspend all non-immigrant visa services, taken on Sunday in response to last week's arrest.
On Monday Turkey's Foreign Ministry said the suspension was causing "unnecessary tensions".
Turkey has also suspended "all visa services".
Several Turkish news networks announced on Monday that a warrant had been issued for a second consulate worker.
They said the suspect was still being sought but his wife and child were being questioned.
Last week consular employee and Turkish citizen Metin Topuz was arrested, accused of espionage and links to US-based cleric Fethullah Gulen.
Ankara has for months been pressing Washington to extradite Mr Gulen over his alleged role in the botched coup in July 2016.
Turkish President Recep Tayyip Erdogan accuses Mr Gulen of instigating the unrest - a charge the cleric denies.
In the aftermath of the coup attempt, which was led by military officers, 40,000 people were arrested and 120,000 sacked or suspended.
The new diplomatic low between the US and Turkey comes less than a month after Donald Trump said ties between the countries were "close as we've ever been".