The gulf nation's cabinet is currently appointed by Bahrain's king, with about half the members coming from the royal family.
Sunni politicians in the parliament criticised the al-Wefaq demand made on Thursday, with the starkest criticism coming from the al-Menbar society.
|Shia-Sunni relations in Bahrain have often been strained in recent years [File: AFP]
"The current constitution is a national achievement that must not be compromised; it is the legal reference for [relations] between the rulers and the ruled and it needs to be respected and cannot be violated except through the constitutional framework," the al-Menbar group said in a statement.
But Marzooq said he was not criticising the king. "We're not challenging the al-Khalifa family, we deny this. Nothing has been said against this family or any other," he said.
Nevertheless, Shia demands added a diplomatic dimension this week when Adel al-Mouawda, one of eight deputies in the Sunni Al Wasalah bloc, warned the British embassy in the capital Manama against meddling in Bahrain's internal affairs.
The British ambassador to Bahrain had met with Al Wefaq's parliamentary bloc a few days after its annual assembly.
"His meeting with Al Wefaq bloc is a clear abuse of the diplomatic relations between member states of the United Nations," Mouawda was quoted as saying by the official Bahrain News Agency (BNA).
He said such meetings were "against the unity and cohesion of the Bahraini people". The British mission in Manama denied it was interfering in Bahrain's internal affairs.
King Hamad bin Isa al-Khalifa started a reform process in the early 2000s that led to a new constitution and parliamentary elections in 2002 and 2006.
Bahrain's next parliamentary elections are expected in November, and observers say the debate on the constitution is part of the bitter wrangling between Shia and Sunni blocs ahead of the polls.
The country's elected parliament is the only one in the Gulf Arab region besides Kuwait's assembly, but its bills still need to pass an upper house that is appointed by the king.