|Poland's lower house of parliament|
The Palikot Movement argues that the cross breaches Poland's secular constitution.
The idea is opposed by conservatives, who believe that the state should defend religion.
The role of the Church in public life remains an important symbolic issue in Polish politics.
The disputed crucifix currently hangs in the Sejm, the lower house.
According to a formal petition submitted by Palikot, the crucifix contradicts Poland's officially secular constitution, and should be removed.
In a letter to the speaker of the parliament, Palikot argues: "The Republic of Poland is a secular state whose authorities ... should remain impartial on religion and philosophical matters."
End Quote Jaroslaw Kaczynski Former Polish prime minister
'Clash of cultures'
Conservatives have been quick to oppose the idea. Poland's largest opposition party, the conservative Law and Justice, has submitted a counter-proposal to keep the cross in parliament.
Its leader, former Prime Minister Jaroslaw Kaczynski, described the issue as a "clash of cultures" and argued that it raised "questions not only of faith and religion [...] but also of patriotism and national pride".
For both liberals and conservatives, the influence of Catholicism on Polish society remains an important issue.
The Palikot Movement has established itself as a secular and liberal voice in Polish politics, and is opposed to the Church's position on social issues, including gay rights and abortion. Poland's only transsexual member of parliament is a Palikot member.
In elections last month, Palikot emerged as Poland's third-largest parliamentary party.
Correspondents say conservative Poles are more sympathetic to the Church, with many remembering the role that it played in opposing Communism during the Cold War.
Many Poles remain sensitive about public displays of faith. Last year, a proposal to remove a cross commemorating former President Lech Kaczynski, who was killed in a plane crash in Russia last year, led to rioting in the centre of Warsaw.