Meanwhile, the prime minister has warned against street protests. Turkey's main opposition party has filed a formal request seeking Sun.
The pro-Kurdish opposition HDP, which had appealed along with the CHP for the referendum to be annulled, said the late decision by the electoral board to allow unstamped ballots meant it was impossible to determine how many invalid or fake votes may have been counted.
"We can not stay silent over regime attacks anymore", Trump told Erdogan, the source said.
On Monday, Kremlin spokesman Dmitry Peskov told reporters that the results of Turkish referendum should be respected.
Electoral board head Sadi Guven said the objections would be evaluated Wednesday.
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". Two opposition parties challenged the outcome with the electoral board, but it rejected the motions earlier in the day.
Both Turkey's bar association and global election monitors from the Organisation for Security and Cooperation in Europe (OSCE) said that the illegal move may have swung the knife-edge vote, which will give Mr Erdogan the power to appoint and fire ministers, name half the members of the country's highest judicial body, hold the leadership of a political party while in office, and possibly stay in power until 2029.More news: Trump extends private-sector health care program for vets
The U.S. response has been different, with President Donald Trump calling Erdogan shortly after the referendum to congratulate him on his win.
The constitutional changes move Turkey from a parliamentary system of government to a presidential one.
"The main opposition party not recognizing the results is not an acceptable thing", Yildirim said.
Observers from the Organization for Security and Cooperation in Europe (OSCE) and the Council of Europe said Sunday's referendum had been an uneven contest. "We will follow closely how Turkey behaves on this".
A defiant Erdogan has dismissed complaints from his domestic opponents and European observers, saying the vote put to an end all debate over the powerful presidency he has long sought.
Meanwhile, Foreign Minister Mevlut Cavusoglu has said that a critical report by European observers on the referendum contained several mistakes which he believed were deliberate.