After a morning of talks between an EU team led by Tusk and Turkish Prime Minister Ahmet Davutoglu, the source said Tusk had reconvened an EU summit to endorse the results.
"Tusk is ready to put a text on the table for all EU leaders and recommend its adoption based on negotiations with Turkey overnight and today," the source added.
A senior EU official said Davutoglu had indicated Ankara would accept the proposal if the EU leaders approved it.
Under the pact, Ankara would take back all illegal migrants who cross to Greece, including Syrians, in return for the EU taking in thousands of Syrian refugees directly from Turkey and rewarding it with more money, early visa-free travel and progress in its EU membership negotiations.
Czech Prime Minister Bohuslav Sobotka tweeted from inside the room: "There is a strong will to conclude the deal."
Migrants who arrive in Greece from next Monday will be subject to being sent back to Turkey once they are registered and their asylum claim is processed, an EU official said.
While the talks were underway, Turkish President Tayyip Erdogan accused the EU of hypocrisy over migrants, human rights and terrorism after a few dozen supporters of the separatist Kurdistan Workers Party (PKK) set up protest tents near the EU summit center.
Erdogan said Europe was "dancing in a minefield" by directly or indirectly supporting terrorist groups.
Europeans need to look at their own record on migrants before telling Turkey what to do, he said, adding that Ankara would only listen to EU criticism on rights when it was correct.
Note EU-Digest: Most political pundits in Turkey are saying that Mr. Erdogan, having totally suppressed the free press in his country, angered just about every human rights group, proven to be involved in numerous corruption deals, and who has lost most, if not all support abroad and in his own country, is only able to stay in power for lack of a strong opposition and the passiveness of its population As a US reporter noted: "the ghost of Ataturk can't help the Turks to topple Erdogan, they will need to become far more proactive themselves".
Read more: EU and Turkey Reach Deal to Curb Migrant Flow Amid Erdogan's Hypocrisy Accusations