Ruling party MPs resolved on Friday to defend their disputed electoral victories in parliamentary elections held last June, chief parliamentary whip Joram Gumbo said.
The move comes a week after Zanu-PF forced the country's chief justice to take early retirement.
Government critics feel that in the face of a serious threat from the opposition Movement for Democratic Change, the government it is trying to tighten its grip on those institutions which have demonstrated their independence.
The official Herald newspaper quoted Zanu-PF spokesman Jonathan Moyo as saying a meeting of its members of parliament on Friday passed a vote of no confidence in the country's highest court of appeal and urged justices Nicholas McNally and Ahmed Ebrahim to resign.
Justice McNally said Justice Minister Patrick Chinamasa asked for his resignation and that of his colleague Justice Ibrahim.
"We were told very nicely and politely we should take our leave and go, otherwise anything could happen. They didn't want me to come to any harm," he said.
But Justice McNally, who is white, has turned down the request while Justice Ebrahim, of Asian origin, has indicated he wants to consult his family, the paper said.
Sternford Moyo, president of the Law Society, told the BBC that if the judges agreed to resign in such circumstances it would set a precedent, making it difficult for the courts to rule against the government.
He also praised Justice McNally for refusing to step down and urged the other judges to follow his lead.
Neither the president nor parliament has the power to sack a judge outright.
Threats
A weeks ago, Anthony Gubbay, the chief justice, was forced to agree to retire after being told by the government that decisions had been incorrect and his safety could not be guaranteed in the wake of threats from ruling party militants.
The Supreme Court ruled on 30 January that the government decree banning opposition challenges to results of last June's parliamentary election infringed candidates' constitutional rights to seek legal recourse.
Ruling party MPs resolved on Friday to defend their disputed electoral victories in parliamentary elections held last June, chief parliamentary whip Joram Gumbo said.
The Supreme Court comprises two whites, one Asian and two blacks, but the government says its rulings generally reflect a "Rhodesian mentality".
Rhodesia was Zimbabwe's colonial name under white minority rule before Mr Mugabe's party took power after independence from Britain in 1980.
In December, the court effectively declared unlawful Mr Mugabe's drive to seize white-owned farms for black peasants without paying compensation.
Hundreds of militants stormed the supreme court in Harare on 24 November, disrupting a hearing on land seizures. No one was arrested for the disturbance.
Mr Mugabe has repeatedly vowed the courts would not stop his plans to seize more than half the 12 million hectares (30 million acres) of farmland occupied by 4,500 white farmers for blacks to redress what he calls the imbalances caused by Zimbabwe's colonial past.
Meanwhile, hundreds of armed riot police have forced journalists to abandon a protest against intimidation of the independent press and an anonymous bomb attack that damaged the press of the Daily News, a newspaper the government had accused of opposition bias.