Tsai and Lai, who is widely expected to stand for the presidency in elections due next year, "hope to do their part to help Turkey rebuild its homeland as soon as possible", the presidential office said in a statement.
Turkey, like most countries, has no diplomatic relations with Chinese-claimed Taiwan, but the two maintain de facto embassies in each other's capitals and there are direct flights between Istanbul and Taipei.
Tsai visited the de facto Turkish embassy in Taipei on Thursday to sign a book of condolence, writing: "My heart goes with our Turkish friends. Taiwan stands with Türkiye!", according to her office.
Last year, Tsai and Lai also both donated one month's salary to aid humanitarian relief efforts for war-torn Ukraine. Tsai earns around T$400,000 ($13,300) a month.
The death toll from the earthquake and aftershocks earlier this week passed 15,000 on Thursday amid anger from those left destitute and frustrated over the slow arrival of rescue teams.
Taiwan, which frequently suffers earthquakes itself, has already announced $2 million in disaster relief and has sent two rescue teams to Turkey to help in search efforts for survivors.
Tsai spoke by video call to some of Taiwan's team on the ground on Wednesday.
"I would like to thank all the members for not being afraid of difficulties and going all out so that Taiwan and Turkey can help each other," she wrote on her Facebook page.
Turkey joined international efforts and sent rescuers to Taiwan in 1999 after a massive earthquake killed more than 2,000 people.
($1 = 30.0550 Taiwan dollars)
(Reporting by Ben Blanchard; Editing by Lincoln Feast.)
