More than three-quarters of the Armenian population of 120,000 had fled by Friday after a lightning defeat by Azerbaijani forces. The enclave had broken away in the 1990s.
Cyprus traditionally has close ties with Armenia, and has a minority Armenian Christian population represented in parliament.
"The Cypriot government maintains an open corridor for the Armenian people and in that framework is ready to offer immediate humanitarian aid," the Cypriot Foreign Ministry said in a statement.
"Cyprus is considering, among other things, ways to host a number of displaced Armenians in our country should that be deemed necessary."
There have been Armenians in Cyprus for centuries. Many trace their roots back to Armenian people or orphans forced to flee mass killings under the Ottoman Empire in 1915, which some governments today consider genocide.
Turkey accepts that many Armenians were killed in clashes with Ottoman forces during World War One, but denies that the killings were systematically orchestrated or constitute a genocide.
(Reporting By Michele Kambas; Editing by Kevin Liffey)
