Prominent Chinese dissident writer Liao Yiwu told AFP that Liu's wife sent a formal request to China's state security ministry requesting permission for the couple and her brother to leave the country.
The Nobel Peace Prize winner, who was sentenced to 11 years in prison in 2009 for "subversion" after calling for democratic reforms, was released on medical parole after being diagnosed with terminal liver cancer last month, his lawyer said this week.
His wife, Liu Xia, sent the request before the diagnosis, but family friends say the couple wants the 61-year-old democracy campaigner to be treated abroad.
Another friend, who requested anonymity out of fear of persecution, told AFP he had received similar information from family sources.
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Amnesty International China researcher Patrick Poon, citing people close to the family, said: "Liu Xia indeed wants Liu Xiaobo to get medical care abroad." But Amnesty was unable to verify if Liu Xiaobo himself has expressed those wishes.
The state security ministry could not be reached for comment as its phone number is not publicly available.
"I am sick of my life, this grotesque life ... I long to escape," Liu Xia, who has suffered from heart problems and depression, writes in the missive, which Liao posted online.
"I can hardly believe that Xiaobo agreed to leave China together with me and (my brother). I am grateful to you and to our friends for everything you've been doing and cannot wait to embrace you," it added.
Liao said he sent the letter to the US and German governments.
The new US ambassador to Beijing, Terry Branstad, said on Wednesday that he would like to see Liu have the option of treatment abroad, echoing a growing chorus of Chinese and foreign human rights activists.
Authorities in the northeastern city of Shenyang, where Liu is being treated, said late Wednesday that Liu was taken to a hospital after he was found to be unwell on May 31.
On June 7, cancer experts at China Medical University No 1 Hospital determined that Liu had "liver cancer with systemic metastasis", meaning it has spread to the rest of his body, the Shenyang legal bureau said in a statement.
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