But both sides - with a joint interest in improving security cooperation and economic relations - will also not want the talks to fail.
President Recep Tayyip Erdogan on Monday insisted that Turkey still aimed to become a full member of the European Union, hours ahead of a crunch summit with the bloc's chiefs aimed at restoring damaged ties.
European Commission President Jean-Claude Juncker said he was "saddened" by the state of relations with Turkey and urged Erdogan to improve relations with Greece and Cyprus.
"We didn't achieve any kind of concrete compromise today", Tusk told a news conference held with Erdogan.
"What can I say is that I raised all our concerns".
Juncker insisted that, in order to proceed with is European Union bid, Turkey need to uphold democratic values and the rule of law, as well as fix its bilateral ties with neighbouring Greece and Cyprus.
"Only progress on these issues will allow us to improve EU-Turkey relations, including the accession process", he said adding: "We agreed that we should continue our dialogue in hard circumstances".
Tusk said that the Varna meeting had "demonstrated that while our relationship is going through hard times, in areas where we do cooperate, we cooperate well". Scores of civil society groups, media outlets and television stations have been closed. But Erdogan added: "We are going to remind them (Tusk and Juncker) once more that Turkey will not tolerate hypocrisy".
The Turkish leader also extended his honest condolences of the Kemerovo tragedy and offered the words of sympathy to the families of those killed in the disastrous shopping mall fire.
Erdogan complained that the bloc has so far disbursed just €1.8 billion of funds for the care of Syrian refugees and dragged its feet on a pledge to end visa requirements for Turks travelling to Europe.
Before arriving in Bulgaria, Mr Erdogan insisted "EU membership remains our strategic goal".
"We also wanted to show the truth to the Iraqi authorities, who reject Turkey's assertions about the PKK presence in the district", he said.
Turkey, for its part, has accused Brussels of failing to show solidarity after the coup and appears set on forging a strong partnership with President Vladimir Putin's Russian Federation.
"No one expected a breakthrough now, but neither side wants a break-up", Zeynep Alemdar, chairwoman of the global relations department at Okan university in Istanbul, told the Financial Times.
"I would like to express our appreciation for the impressive work Turkey has been doing, and to sincerely thank Turkey and the Turkish people for hosting more than 3 million Syrian refugees these past years", he said.