The two countries, which have no diplomatic ties and a history of animosity stemming from the mass killings from both sides during World War One, announced late on Monday they would sign accords within six weeks under a plan to end a century of hostility.
"If everything goes as planned, if mutual steps are taken, the borders could be opened around New Year," Davutoğlu told Turkish NTV television during a visit to northern Cyprus.
The plan to normalise ties was announced in April, but Monday's statement marked the first real progress.
Under the agreement, both sides would hold domestic consultations before signing two protocols on the establishment of diplomatic relations and development of bilateral relations.
The protocols would have to be ratified by parliaments of the two countries.
In his first comment on the issue, Prime Minister Tayyip Erdoğan stressed that the accords would not go into effect until the Turkish parliament had approved them.
Reopening the border and establishing ties with Armenia would increase predominantly Muslim Turkey's influence in the region and aid its faltering bid to join the European Union.
The bloc has long asked candidate member Turkey to normalise ties with its neighbour and restoring diplomatic ties would also be beneficial for regional security.
The EU on Tuesday welcomed steps to establish bilateral ties and urged the two countries to implement the protocols rapidly.
"This agreement should contribute to peace and stability in the South Caucasus," EU commissioner for external relations Benita Ferrero-Waldner and EU enlargement commissioner Olli Rehn said in a statement.
Diplomatic ties would also give landlocked Armenia, reeling from the global financial crisis, access to Turkish and European markets.
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