The developments came barely hours after the largest single entity, Bharatiya Janata Party, conceded that it was not in a position to form the government despite its pre-poll alliance having a "clear public mandate".
Koshyari called upon the Sena leader Eknath Shinde to convey his stand in the matter to him.
Though a specific time-frame is not indicated in the Raj Bhavan communique, it is learnt that the Sena, which has 56 members in the 288-member house, may have barely 24 hours to stake its claim along with the requisite number of supporting MLAs before the Governor.
As per indications, the Sena is likely to secure the outside support of the third largest party - Nationalist Congress Party (55 MLAs), and the fourth-largest Congress (44) - in its bid to form the government.
Both the NCP and the Congress have made it adequately clear that their support would depend on the Sena categorically announcing that its alliance with the BJP has broken and then withdraws its sole nominee in the Union Cabinet, Heavy Industry Minister Arvind Sawant.
Meanwhile, in the wake of the Governor's invite, a fresh round of informal negotiations have started between the Sena-NCP-Congress late this evening.
The caller, whose identity is not clear, reportedly called up at least twice on the landlines of "Matoshri" - the CM's private residence in Bandra east - on Saturday and said he wanted to speak to Thackeray, said Transport Minister Anil Parab.
"We have immediately informed the Mumbai Police to do whatever is necessary in the matter. The police will verify the antecedents of the caller and take whatever measures necessary for the safety and security of the CM," he told media persons this evening.
However, Parab categorically refuted reports in some sections that the caller allegedly issued threats to blow up the CM's residence, saying that there was no such thing.
Home Minister Anil Deshmukh said that the government has taken strong note of the development and the CID has been directed to look into the matter which has come up on the eve of the Monsoon Session of Maharashtra Legislature starting here on Monday.
Meanwhile, leaders of the ruling coalition - the Shiv Sena, the Nationalist Congress Party and the Congress - including Eknath Shinde, Parab, Balasaheb Thorat and Ashok Chavan, besides Mumbai Mayor Kishori Pednekar and others have reacted strongly to the incident and said the government will not be cowed down by any such threats to the CM.
All leaders have expressed full support to Thackeray while Pednekar said Shiv Sainiks will be ready to deploy their own ring of security around "Matoshri" if needed.
Taking serious note of the incident, the Mumbai Police have beefed up security around the Thackeray home as well as the Vidhan Bhavan complex at Nariman Point as a precaution
(IANS)