The company is thought to require at least £210m in short-term funding to continue operating and implement a turnaround plan.
They added: "Based on our current trading assumptions (Dec' 18 EBITDA £188m with material downside risk) and our estimates of the mounting debt ( £1.1bn), likely additional supply chain funding/working capital unwind ( £300m) and pension liabilities (£600m), we now see no equity value".
The general secretary of the RMT rail union, Mick Cash, said Carillion's workers were "not responsible for the crisis".
The meeting comes as Carillion - a major government contractor on projects including High Speed 2 - races to secure additional funding to avoid collapse, with around 19,500 United Kingdom jobs at risk.
Unite represents more than 1,000 workers at the company and has many more members in the supply chain.
Yesterday, government officials met to consider the company's precarious position, while today pension trustees and regulators convened to discuss the firm's £587m pension shortfall. "It would be inappropriate for us to comment on any individual contractor's internal financial governance".
The rescue plan shown to lenders on Wednesday includes handing back some loss-making contracts, revising the terms of others and potentially accepting financial support from the Government if it can not secure it from private sector sources.
To date, numerous banks have indicated that they are reluctant to provide additional funding given the potential for huge losses on their existing exposure.
The person familiar with the situation, who asked not to be named due to the sensitivity of the situation, said the government needed to get involved to help prop up the supplier.
Its only asset sale since the crisis erupted has been to offload a portfolio of healthcare contracts to rival outsourcer Serco for £50m - against a broader forecast for disposal proceeds of £300m.
One of many private companies to run public services in Britain, Carillion is fighting to survive after costly contract delays and a downturn in new business prompted a string of profit warnings and a first-half loss of more than 1 billion pounds ($1.4 billion).
Such a plan would leave its pension scheme, or the Pension Protection Fund, as a big shareholder.
The company is the UK's second-largest construction company and a key government contractor.
Its share price plunge more than 70% in the past six months after a shock profit warning wiped nearly £600m from the company's stock market value in July previous year.
Since then, the company has cleared out its executive team, including chief executive Richard Howson and finance director Zafar Khan.
Carillion's fight for survival is being led by interim boss and industry veteran Keith Cochrane, a former CEO of engineer Weir Group. "We are committed to maintaining a healthy supplier market and working closely with our key suppliers".
"The company has kept us informed of the steps it is taking to restructure the business".
A spokeswoman for the government declined to comment to the BBC on any specific meetings.
"We remain supportive of their ongoing discussions with their stakeholders and await future updates on their progress".