The phone conversation between Tillerson and Cavusoglu was the first since the diplomatic crisis between the two North Atlantic Treaty Organisation allies broke out.
"A decision between the two countries has been made for representatives to come together and work on this issue", he said in an interview on broadcaster Haber Turk.
Turkish Deputy Prime Minister Mehmet Simsek said on a visit to Washington that Turkey is "safeguarding" the USA employees and "we don't want this dispute to last more than a second".
The U.S. Embassy on Sunday announced the suspension of some visas for Turkish nationals following the arrest of a Turkish employee at the U.S. Consulate in Istanbul, prompting a tit-for-tat response from Ankara.
On Wednesday, a different Turkish Deputy Prime Minister, Mehmet Simsek, said the crisis between the two countries was "overblown" and will likely be resolved soon.
"Relations between Turkey and the USA are stable", he said.
Among them is also an American pastor, Andrew Brunson, who was detained last October and has been held in a Turkish prison ever since, despite repeated calls for his release.
Turkey's ties with the USA have frayed since the failed coup against Erdogan in July 2016.
Speaking to reporters in Ankara on Wednesday, US Ambassador John Bass said the US sought an official explanation from Ankara concerning the arrests.
Atilla was arrested in March and indicted in September along with Caglayan, who is in Turkey.
Secretary of State Rex Tillerson told Turkey's foreign minister that the Trump administration is profoundly "concerned" about the detention of USA citizens and the arrest of local Turkish consular employees, the State Department said.
The phone call marks the first time the two ministers spoke since the visa dispute over Turkey's detention of a USA consulate staff for links to the Gülenist Terror Group (FETÖ), which is responsible for the July 15 failed coup attempt.