The offer came a day after China rejected criticism over its treatment of Liu, as the United States urged Beijing to give the paroled activist freedom to move and choose his own doctors.
Liu's lawyer Mo Shaoping revealed on Monday that the 2010 Nobel Peace Prize laureate had been hospitalised after being diagnosed with late-stage liver cancer in May.
The writer, now 61, was sentenced to 11 years in prison in 2009 for "subversion" after spearheading a bold petition for democratic reforms.
Taiwan said it would "welcome" Liu for treatment at a time when relations with Beijing have worsened dramatically under China-sceptic President Tsai Ing-wen.
China still sees Taiwan as part of its territory to be reunified.
But the self-ruling island has developed its own democratic political system and has become home to some activists fleeing China.
"We urge Beijing to immediately release Liu and let him choose wherever he wants to receive treatment," said Chiu Chui-cheng of the Mainland Affairs Council, Taiwan's top policy-making body on China.
Chinese dissident Wang Dan, a Tiananmen protest leader who lives in exile in Taiwan, said that he has contacted Germany's foreign ministry in the hope the country would take Liu for treatment, as it has a hospital famous for treating liver cancer.
The new US ambassador to China, Terry Branstad, said today he would like to see Liu treated abroad.
Human rights groups have also called on Chinese authorities to give Liu the chance to seek treatment elsewhere.
Beijing has stepped up pressure on her government, from luring away the island's diplomatic allies to arresting a Taiwanese activist on "subversion" charges on the mainland.
China has cut off all official communication with Taipei and has upped military drills while Taiwan is developing home-grown defence systems in response to what it sees as a growing threat.
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