Speaking after talks with President Recep Tayyip Erdogan in Ankara, May said she was "proud" Britain had stood with Turkey's democratically-elected government when the coup struck.
"Now it is important that Turkey sustains that democracy by maintaining the rule of law and upholding its international human rights obligations, as the government has undertaken to do," she said.
May stressed that Turkey was one of Britain's "oldest friends" with relations going back over 400 years, referring to the establishment of relations between the Ottoman Empire and England under Elizabeth I.
May said Britain and Turkey had agreed to set up a working group to boost bilateral trade.
"We have agreed that we will have a joint working group to prepare the ground for our post-Brexit trading relationship," May said.
Erdogan said the two sides were targeting increasing annual trade from over $15 billion to $20 billion.
May said that they had discussed the prospects for enhancing trade in the talks.
Erdogan added: "We discussed what kind of joint steps we could take in the defence industry."
The pair also discussed the battle against jihadists in Syria and efforts to reunify Cyprus, where both Ankara and London are guarantor powers.
The British premier said security on air transport was also a key issue.
"We are also agreed to step up our work together on aviation security with a programme of shared training and information exchange," said May.
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