Trade, military, economic and cultural relations, were on the agenda of the two leaders during a meeting held in Ankara.
Türkiye has said it was ready to assist Hungary on natural gas delivery via the Trans Anatolian Natural Gas Pipeline Project (TANAP).
"We, (Türkiye) together with Azerbaijan, are ready to provide all means of support in the delivery of natural gas to Hungary via TANAP," President Recep Tayyip Erdogan said at a joint press conference on Wednesday with his visiting Hungarian counterpart Katalin Novak at the presidential complex in the capital Ankara.
Hailing the longstanding ties between the two countries, Erdogan said the "helping hand" Hungary extended Türkiye after twin devastating earthquakes in February was the "solid example" of these strong relations.
The 7.7 and 7.6 magnitude earthquakes struck 11 Turkish provinces — Adana, Adiyaman, Diyarbakir, Elazig, Hatay, Gaziantep, Kahramanmaras, Kilis, Malatya, Osmaniye, and Sanliurfa, claiming more than 50,000 lives.
Over 13.5 million people in Türkiye have been affected by the quakes, as well as many others in northern Syria.
"We discussed various aspects of Türkiye-Hungary relations at the meetings we held today," he added.
Trade, military, economic and cultural relations, were also on the agenda in the meeting between the two leaders, during which they watched footage from the earthquake, Erdogan noted.
They also affirmed their common will to further strengthen ties based on an enhanced strategic partnership, Erdogan said, adding that the two countries agreed to hold a sixth meeting of their High Level Strategic Cooperation Council in the Hungarian capital Budapest in December.
READ MORE: 'Race against time': How Hungarian team saved 17 people from Türkiye quakes
Expanding defence co-operation
Erdogan said that he and Novak emphasised the importance of developing co-operation between the two NATO allies in the defence industry.
Asked about Russia's war on Ukraine, President Erdogan said Ankara wanted to bring Moscow and Kiev to start negotiations and "end the war with peace at the table."
For her part, President Novak hailed Türkiye's role in the Black Sea grain initiative and urged an immediate cease-fire between Russia and Ukraine.
She further noted that Türkiye was "indispensable for Hungary's energy security" and noted that her country needed the Turkish Stream.
On NATO, Novak said her country favored enlargement of the alliance and called for steps to be taken in this direction.
Pointing to the Hungarian parliament's recent approval of Finland's bid to join NATO, Novak said: "A positive decision has been made in Hungary regarding Finland joining (the alliance), and the issue of Swedish accession is on the agenda and discussions are ongoing in the Hungarian parliament."
Abandoning decades of military non-alignment, Finland and Sweden formally applied to join NATO last May.
But Türkiye, a longstanding NATO member, asked the two Nordic countries to take concrete action against terrorist groups like the PKK and the Fetullah Terrorist Organization (FETO).
In June, Finland and Sweden signed a memorandum with Türkiye to address Ankara’s security concerns, and senior diplomats and officials from the three countries have held various meetings since then to discuss the implementation of the trilateral agreement.
Sweden passed an anti-terror law last November, hoping that Ankara would approve Stockholm’s bid to join NATO. The new law, which will go into force on June 1, will allow Swedish authorities to prosecute individuals who support terrorist organisations. Türkiye says the adopted laws were not sufficient enough, and nothing much had been done to stop the activities of the terror groups.
Türkiye said earlier this month that it would approve the process of Finland's NATO membership protocol in parliament.
READ MORE: Only Türkiye has successfully mediated between Russia, Ukraine: Hungary