The party said it can prove its majority within seven days of swearing-in of its chief minister.
"We undertake to prove the majority on the floor of the House within seven days of swearing in. Kindly initiate the process of inviting us to form the government at the earliest," Kavlekar said in the letter submitted to Sinha.
In the Assembly elections held in February last year, the Congress had bagged 17 seats in the 40-member House, falling short of the majority figure by four seats.
The BJP, which had bagged 13 seats, firmed up an alliance with Goa Forward Party and Maharashtrawadi Gomantak Party, both of which had secured three seats each, and also took support of three Independents.
Subsequently, the Governor invited the BJP-led coalition to form the government. The present dispensation is headed by Manohar Parrikar, who is currently undergoing treatment in the US for a pancreatic ailment.
"Despite the Congress being the single largest party, Manohar Parrikar was invited to be the chief minister. He was administered the oath of office and secrecy and the government headed by him was constituted," Kavlekar added in the letter.
"Your Excellency is aware that Indian National Congress, even as on date, is the single largest party in the state Legislative Assembly having 16 elected representatives," it added.
The Congress also referred to the developments in Karnataka, where Governor Vajubhai Vala invited the BJP to form the government after the Assembly election results declared on Wednesday.
The BJP, which emerged as the single largest party with 104 seats in the 224-member Karnataka Assembly, was invited by the governor despite the JD(S)-Congress combine staking claim by citing the support of 117 MLAs. The majority mark is 112 seats as polls in two constituencies were deferred by the Election Commission.
"In Karnataka, in the recently-concluded elections, the Congress and the Janata Dal (Secular), which has more than the required numbers, staked claim to form the government.
"However, the Governor of Karnataka has invited B S Yeddyurappa of the BJP to form the government apparently on the precedent that the single largest party is required to be invited to form the government. This shows that the decision taken in 2017 in Goa was patently wrong," the letter said.
"Your Excellency is, therefore, urged to dismiss the present illegitimate government in the state and invite the Leader of Indian National Congress Legislature Party to form the government," the letter said.
Speaking to reporters after meeting, Chellakumar said the party has requested the governor to revert to the letter in seven days. "If she does not get back, then we will decide our future course of action. We have the numbers with us," he said.
Although the Congress had won 17 seats in Goa last year, it currently has 16 members in the Assembly as one of its legislators, Vishwajeet Rane, had joined the BJP last year. Rane later won Valpoi Assembly bypoll on a BJP ticket.
"We have all the required numbers with us to form the government in the state. We can prove our numbers on the floor of the House. The magic figure of 21 is there with Congress... even without poaching into the BJP camp," Chodankar said.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)