A brief readout of Monday's (17 April) phone call, released by the White House, did not say whether Trump had raised concerns expressed by worldwide observers monitoring Sunday's referendum over voting irregularities.
The Republican People's Party (CHP) said on Tuesday it has filed paperwork requesting that the narrow 51.3 per cent vote in favour of constitutional reform that will make President Recep Tayyip Erdogan much more powerful be annulled because of voting irregularities which made it "illegitimate".
Opposition parties have complained of a series of irregularities, particularly an electoral board decision to accept ballots without official stamps, as required by Turkish law.
The German government has rejected calls from opposition parties to suspend Turkey's European Union membership talks after Sunday's referendum in which more than 50 percent of Turks voted "Yes" to a transition to a presidential system.
Foreign Minister Mevlut Cavusoglu said that Erdogan and Trump would meet in person next month, before a North Atlantic Treaty Organisation summit. This work has now finished. A New York Times editorial said the referendum has left Turkey "in the hands of an erratic and vengeful man and the world wondering whether a nation that for decades has served as a crucial bridge between Europe and the Muslim world can possibly have a stable and prosperous future under someone with so little respect for democratic structures and values".
The differing statements underscored the continuing difficulty facing the Trump administration in managing its foreign policy messaging.
Hurriyet said around 14 protesters were detained by the police in the Mediterranean province of Antalya, where a majority of people voted against the constitutional changes in Sunday's referendum.
Sunday's referendum allows Erdogan, who has ruled Turkey first as prime minister and now as president since 2003, to fulfill his long-held ambition for a presidency with executive powers.
On Tuesday, Yildirim said Erdogan would be invited to join the party as soon as the official results are declared.
Turkey's "yes" campaign was led by the governing AK Party and supported by the right-wing MHP.
"When the High Electoral Board [YSK] announces official results, our president will be able to return to the party", Yildirim told reporters in front of AKP headquarters.
Turkey's three largest cities - Istanbul, Ankara and Izmir - all voted "no" although "yes" prevailed in the country's Anatolian heartland.
The White House emphasised that the two leaders were united in their determination to punish Syria's president, Bashar al-Assad, for using chemical weapons against his own people. "We will follow closely how Turkey behaves on this".
European leaders adopted a similar tone, with German Chancellor Angela Merkel cautioning that the "tight referendum result shows how deeply divided Turkish society is and that means a big responsibility for the Turkish leadership and for President Erdogan personally".
The two leaders also discussed Turkey's support for the USA response to a Syrian chemical weapons attack and efforts to counter the Islamic State group, according to a White House summary of their phone call Monday.