Eide criticised the government for being far too passive in its stance over the treatment of Liu, who died on Thursday at the age of 61.
The PM is currently on holiday and announced her “deep grief” at Liu's death, reports NRK.
Eide called the reaction of Solberg “ill-mannered and quite provocative”.
“I think it is ill-mannered and quite provocative that the prime minister uses words such as ‘deep grief' when she heard about Liu Xiaobo's death,” said Eide, who is standing as a candidate with the Socialist Left Party at this year's general election.
“Solberg in all likelihood reacted in the opposite manner – with relief,” Eide told broadcaster NRK.
Liu, a former figurehead of the 1989 democratic movement of Tiananmen Square, was honoured with the Nobel peace prize in 2010 for "his long nonviolent struggle for fundamental human rights in China".
Even though the Norwegian Nobel Committee is independent of the government, Beijing froze diplomatic ties with Oslo as a result of Liu winning the peace prize.
Beijing and Oslo relations were normalised last December.
Solberg has come in for criticism for declining to speak about Liu after he was moved from prisoned to a secured hospital ward due his deteriorating condition.
The PM responded to Eide's criticism via a statement released via the state secretary Ingvild Stub.
“We must respect that there are different opinions in politics, and that the glass ceiling in Norwegian politics is generally high. Eide's claims that the prime minister felt relief over a death are as sad as they are unfair, but can unfortunately probably be linked to the fact that there is an election this year,” said the statement.
Eide told NRK that the government's actions during recent weeks showed anything but sympathy for Liu's situation.
“Liu heard nothing from Erna Solberg during the period in which he was sick. No warmth, no recognition. There has only been silence from Norway, and that silence has been a sign of support for the Chinese authorities. She surely feels the opposite of grief,” said the former Amnesty International leader to NRK.
Meanwhile, the former chairman of the Norwegian Nobel Committee on Friday justified awarding Chinese dissident Liu with the 2010 peace prize which Beijing slammed as "blasphemy".
"The struggle for Human Rights is peace building," Thorbjørn Jagland, who is still a member of the Nobel committee, said on Twitter.
"That's why the Committee I chaired awarded Liu Xiaobo the Peace Price," he added.
The struggle for Human Rights is peace building. That's why the Committee I chaired awarded #Liu Xiaobo the Peace Price.— Thorbjørn Jagland (@TJagland) July 14, 2017
Liu was not able to attend the Nobel award ceremony in Oslo in 2010 as he was serving an 11-year prison sentence for allegedly "attempting to undermine political order".
The former head of the Nobel committee placed that year's peace prize on an empty chair to honour Liu.
A Chinese foreign ministry spokesman on Friday said that awarding the Nobel peace prize to Liu "goes against the purposes of this award" and is "a blasphemy".
Contacted by AFP, the Nobel Committee refused to comment. It is unclear whether Jagland's tweet was a reaction to the Chinese statement.
The current leader of the Nobel committee, Berit Reiss-Andersen, who wanted to attend Liu's funeral, was not allowed to apply for a visa to China, Norwegian public broadcaster NRK said.
The Chinese embassy in Oslo explained that a visa could not be granted to meet a deceased person and that it would require an invitation from either Liu's widow or relative, NRK reported.
Following Liu's death on Thursday, the Nobel Committee said China was "bearing a heavy responsibility" for his "premature" death and criticised the fact that he was not able to receive "adequate medical treatment".
Germany and the United States had offered to take him in for medical care.