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.
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.
"I learned two weeks ago that Liu Xiaobo said that if he dies, he wants to die in the West," Liao, a family friend who lives in Germany, said in a phone interview.
Another friend, who requested anonymity out of fear of persecution, told AFP he had received similar information from family sources.
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 the rights watchdog was unable to verify if the Nobel winner laureate himself has expressed those wishes.
The state security ministry could not be reached for comment.
Liao said he also received a handwritten letter from Liu's wife in April in which she says her husband wants to leave China.
"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 US and German embassies in Beijing declined to comment.
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.
Taiwan on Wednesday also urged Beijing to release Liu and offered him medical treatment in a move that could stoke tensions between China and the self-ruled island.
Asked to respond to Taiwan's offer, Chinese foreign ministry spokesman Lu Kang on Thursday said: "What you're talking about is China's internal affairs. It has no concern with China's diplomacy."
"I'm not aware of what you said and it's not a diplomatic question," Lu said.
Authorities in the northeastern city of Shenyang, where Liu is being treated, said late Wednesday that he 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.
Liu's wife and other family members were with him at the hospital and they expressed "satisfaction" with the treatment, the statement said.
Meanwhile, a video showing Liu undergoing medical exams in prison was posted on YouTube by US-based Chinese-language news site Boxun late Wednesday.
In the video, Liu says he is "grateful" for the "really good care" he has received from prison doctors.
Liu is seen playing badminton in a prison yard, getting an ultrasound and undergoing a CT scan.
Speaking with his wife in a prison visiting room divided by glass, Liu tells her that he "had a physical exam, they took blood... it's very good."
Boxun did not specify where the footage came from or where and when it was filmed. The site's editors and prison authorities could not be reached for comment.
Amnesty questioned the timing of the video's release and its source.
"If the video was leaked by authorities, this would show how anxious the Chinese government is to justify its treatment of Liu Xiaobo," Poon said.