"We look to the government of Turkey to protect the fundamental rights and freedoms of all its citizens - regardless of their vote on April 16 - as guaranteed by the Turkish constitution and in accordance with Turkey's worldwide commitments", the State Department said in a statement.
"We will consider appeals on the results of the referendum today", Guven said as quoted by the NTV broadcaster.
Ultimately, if the referendum stands, Turkey will shift from a parliamentary government to one largely controlled by the president-though numerous changes strengthening the president's powers won't take place until after the next election in 2019.
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Turkey's "yes" campaign was led by the governing AK Party and supported by the right-wing MHP.
The opposition is particularly incensed by a last-minute move by the YSK to accept ballot documents in envelopes without an official stamp.
Sunday's referendum campaign was criticized by worldwide watchdogs.
Gianni Pittella, the Italian leader of the socialists in the European Parliament, said that his bloc of MEPs would hold talks on whether to veto visa-free travel for Turks next week.
Once the changes come into effect, after the 2019 elections, they will empower the president to appoint senior officials, judges and members of his Cabinet, and to do away with the office of prime minister.
Erdogan argues that concentration of power in the presidency is needed to prevent instability.
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Other changes are to be implemented sooner, including scrapping a requirement that the president not be a member of any political party.
Election monitors from the Organization for Security and Cooperation in Europe were seen entering the High Electoral Board headquarters in Ankara.
"The campaign rhetoric was tarnished by some officials equating "No" sympathisers with terrorists", added ODIHR mission head Tana de Zulueta.
Asked to comment on Erdogan's rebuke, de Zulueta said: "I don't have an opinion". We are invited by the Turkish authorities to observe.
"This is not some move we've done for the first time", said Guven, speaking to reporters Monday in Ankara.
According to Yunus Ulusoy of the Essen-based Center for Turkey Studies and Integration Research (ZfTI, ) the number of votes from Germany amounted to 0.23 percent of the total, where just over 51 percent of voters agreed to give current Turkish President Recep Tayyip Erdogan sweeping powers.
A Turkish police helicopter with 12 people on board, including police officers and a judge, crashed on Tuesday in eastern Turkey, the local governor's office said.
Anadolu said Putin and Erdogan emphasised the importance of normalising ties, which are still recovering after Turkish jets shot down a Russian war plane over the Syrian border in 2015.
Bilginsoy contributed from Istanbul and Mehmet Guzel contributed from Ankara.