The move came at a time when the BJP high command has sent three senior leaders -- Ram Lal, BL Santhosh and Vinay Puranik -- here to meet the state party leaders and allies to take stock of the political situation.
Parrikar, 62, who is suffering from a pancreatic ailment, is currently admitted in the All India Institute of Medical Sciences (AIIMS) in New Delhi for treatment.
The Congress, which has 16 members in the 40-member state Assembly, submitted a memorandum to the Governor, urging her not to dissolve the Assembly and instead invite the party to form the government, Leader of Opposition Chandrakant Kavlekar said.
The state is being ruled by the BJP-led Alliance.
The BJP has 14 seats in the Assembly, its allies Goa Forward Party and MGP have three each. Three Independents and an MLA from Nationalist Congress Party also support the BJP.
All the 16 Congress MLAs, led by Kavlekar, went to the Raj Bhavan but could not meet the governor as she is out of the state.
"The party urged the governor not to consider dissolution of the state Legislative Assembly, which is a possibility considering internal fighting in the ruling alliance and illness of Parrikar," Kavlekar told PTI.
He said the Congress has support of legislators from other parties and can form the government, if given a chance by governor.
"We will prove our majority on the floor of the House," he said.
Earlier, BJP leader Ram Lal said the Goa government is stable and no demand has been made for a change in the leadership.
He stated this after a meeting with party MLAs, former legislators and core committee members.
He said the BJP's alliance partners - the Goa Forward Party (GFP), the Maharashtrawadi Gomantak Party (MGP) and Independents - conveyed that they will agree to any political decision taken by the saffron party.