Sources say the firm has a "matter of days" as it teeters on the edge of collapse, with £1.5bn of debt and a pension shortfall of £587m.
Carillion now employs around 20,000 workers in the UK.
It reported revenues of more than £5bn in 2016, and is one of the largest construction service multinationals in the UK.
In 2016 the Wolverhampton-based company, which employs 43,000 people globally, had sales of £5.2bn and until July boasted a market capitalisation of nearly £1bn.
Last month, Carillion said it received all necessary consents to defer the test date for both its financial covenants from December 31 to April 30, 2018, and that it was continuing constructive talks with stakeholders on options for cutting net debt and recapitalizing.
Ministers are drawing up plans to take over prisons contracts worth £200m from Carillion, .
A Government spokeswoman said on Friday: "Carillion is a major supplier to the Government, with a number of long-term contracts".
It is also the second largest supplier of maintenance services to Network Rail.
These include a new £745 million Aberdeen bypass and plans to extend platforms at Edinburgh Waverley station to make way for longer electric trains. The Financial Times reported that the meeting was convened by Cabinet Office Minister David Lidington and included Business Secretary Greg Clark, Transport Minister Jo Johnson and Justice Minister Rory Stewart, as well as ministers from the culture, health, education and communities departments and the Foreign Office.
Their attitude is said to have prompted senior figures close to Carillion to discuss the possibility of emergency financial support from the Government.
Further press reports earlier today said the government is being urged to bring Carillion into public control amid fears the construction company could collapse.
A Government spokesperson repeated a statement given to Sky News last weekend which said it was "committed to maintaining a healthy supplier market and work closely with our key suppliers".
The assistant general secretary of the Unite union, Gail Cartmail, said: "The government must consider all options while the future of Carillion hangs in the balance, including bringing contracts back in-house".
The company is fighting net debts of more than £900 million, following a crisis sparked in July a year ago when it issued a profit warning.
Its share price plummeted 90 per cent after announcing its first profit warning in July previous year.
The Pensions Regulator would not comment on whether it was attending specific meetings, but a spokesman said: "We have been and remain closely involved in discussions with Carillion and the trustees of the pension schemes as this situation has unfolded".