The organisation has this morning said that the "significant financial impact of the COVID-19 pandemic" on the Trust - which runs York Art Gallery, York Castle Museum, Yorkshire Museum and Gardens and York St Mary’s - means that it is entering into consultations with staff.
It said that "survival depends on reduction in costs to offset huge losses in income".
More than 70 per cent of the Trust’s income is made from the sale of tickets and visitors, it said, and has "exhausted every type of government aid currently available, negotiated rental agreements and tirelessly sought other ways of reducing our costs".
The closure of York Art Gallery, York Castle Museum and the Yorkshire Museum for four months has meant a £1.5 million loss to date after the pandemic hit its peak season and "even as we begin to gradually reopen, the Trust is forecasting a significant reduction in visitor numbers and income".
Reyahn King, chief executive of York Museums Trust, said: “I am devastated that the huge financial impact of COVID-19 on York Museums Trust has led us to this point.
"York Museums Trust looks after much of York’s important and cherished art, culture and heritage and as custodians it is our duty to ensure that we can continue to do this.
"We have exhausted every type of government aid currently available, negotiated rental agreements and tirelessly sought other ways of reducing our costs, but we are now in a position where we have to make some very difficult decisions. It is only by doing this that the charity will be able to survive and continue to share and care for York’s collections.
"I would like to thank every single person, and organisation, who has donated to York Museums Trust during the past few months.
"These contributions have made a big difference to our continued operations to this point. We can’t thank you enough and we are determined that the charity will continue to fulfil its mission inspiring people, sharing and caring for the very special art and collections, gardens, buildings and heritage sites in York. ”
The Trust - which has been asked how many people may be made redundant - was formed on August 1 2002 as an independent charity to manage the museums and gallery service previously run by City of York Council.
It has said it will do all it can to source alternative funding.
"We are grateful for a recent emergency grant from Arts Council England helping us survive until September 30," it said.
"Details have not yet been announced about the government’s £1.57bn relief package for cultural institutions and the scale of York Museums Trust’s loss of income means the Trust must reduce costs further to survive.
"Yesterday we informed our recognised trade unions of our intention to enter into staff consultations which may lead to some redundancies."
The Trust is appealing for donations at www.yorkmuseumstrust.org.uk