Manohar Parrikar is recuperating at his residence since October 14 after being discharged from AIIMS.
The Maharashtrawadi Gomantak Party (MGP), a BJP ally in Goa, gave an ultimatum on Saturday that the charge of the chief minister's post be given to its minister Sudin Dhavalikar in view of Manohar Parrikar's illness.
The MGP warned that it would contest the Lok Sabha polls and Assembly by-elections in the state against the BJP candidates if the demand was not met.
Mr Parrikar, who is suffering from a pancreatic ailment, is recuperating at his residence since October 14 after being discharged from the All India Institute of Medical Sciences (AIIMS), New Delhi.
Two Assembly seats -- Mandrem and Shiroda -- fell vacant last month after respective MLAs, Dayanand Sopte and Subhash Shirodkar, resigned as legislators and quit the Congress to join the BJP.
"Party's central committee had a detailed discussion on the leadership issue today. Chief Minister Manohar Parrikar is ailing and, because of that, the administration is completely paralysed for the last eight months," MGP president Dipak Dhavalikar said.
"Therefore, to bring in discipline and efficiency to the administration, we demand that the chief minister's charge be given to senior MGP MLA Sudin Dhavalikar as early as possible," he said.
"If the charge is not given immediately, we will contest all upcoming elections, that is (Assembly) by-election and Lok Sabha polls, by fielding our candidates against official candidate of the government," he said.
Sundin Dhavalikar is presently the second-most senior minister in Goa cabinet after Mr Parrikar.
MGP's central committee will meet again next month to review the situation, Dipak Dhavalikar said.
"We will see if the government has taken our warning seriously or has continued with the same situation," he said, adding that he himself might contest the by-election from Shiroda, while the party would support Independent MLA Jit Arolkar in Mandrem.
MGP is part of the BJP-led coalition, which also includes Goa Forward Party and three Independents.