"We need to make a decision that is beyond the ordinary", Mr Erdogan said, adding he hoped Turkish voters would make the "expected" decision.
Cezar Florin Preda, head of the delegation from the Parliamentary Assembly of the Council of Europe that monitored the vote, cited an inadequate legal framework and late changes in ballot counting.
But the global electoral monitor said that while the referendum's technical processes were well-administered, there were key issues related to media coverage and the circumstances under which the vote occurred, such as Turkey's ongoing state of emergency enacted after a failed coup d'etat attempt past year.
"The comment that the referendum was below global standards is unacceptable", the ministry statement said.
Merkel and Gabriel said that Turkey - as an OSCE member and European Union candidate country - needs to consider these concerns.
Preda was joined by Tana de Zulueta, head of another team of worldwide observers.
But after voting was underway, the board said it would accept unstamped ballots "unless they are proven to have been brought from outside".
Turkish President Recep Tayyip Erdogan says he was forced to put a tough fight against the "powerful nations of the world" who opposed his campaign for a yes vote in a referendum to expand the powers of his office.
They have also demanded a recount claiming that there had been voting irregularities.
Giraffe gives birth as millions watch live feed April the giraffe has given birth to a calf in a New York Zoo, as a live internet audience of over a million people watched on.
Erdogan himself struck a defensive note after the referendum saying "we want other countries and organizations to show respect to the decision of our people".
The race was marred by divisive rhetoric, with the government equating no voters with terrorist groups and the opposition accusing the ruling party of seeking to install a dictatorship.
Erdogan showed no signs of being cowed by the narrow margin of victory.
"Political discussions about that need to take place as quickly as possible, both at the bilateral level and between the European institutions and Turkey", Merkel and Gabriel said. Later in Istanbul, he changed his tone.
It would also mean that Erdogan, who became president in 2014, could seek two more five-year terms leaving him in power until 2029.
President Erdogan has insisted the changes are needed to amend the current constitution, which was written by generals following a military coup in 1980, to confront security and political challenges in Turkey and avoid fragile coalition governments of the past.
"There is only one decision to ease the situation in the context of the law - the Supreme Election Board (YSK) should annul the election", the Dogan news agency quoted Bulent Tezcan as saying after the CHP alleged widespread violations in Sunday's poll.
If the opposition failed to support such a bill, he said another referendum could be held on reinstating the death penalty.
Under the changes, most of which will only come into effect after the next elections due in 2019, the president will appoint the cabinet and an undefined number of vice-presidents, and be able to select and remove senior civil servants without parliamentary approval. "So I think it's been very clear that elections are not on the agenda".
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