Sky News has learnt that Mr Crozier, the former boss of ITV and the Football Association, is leaving Vue to focus on a reshuffled portfolio of directorships that includes chairing Asos, the online fashion retailer.
Insiders said the search for his successor had been underway since early January, with an appointment expected imminently.
Mr Crozier's departure comes as the UK and swathes of the global cinema industry contend with the implications of the COVID-19 outbreak, which has forced thousands of multiplex sites to temporarily close.
Vue, which is run by Tim Richards, one of the industry's most successful entrepreneurs, had been contemplating a sale process last autumn, but was hampered by the declining valuations of many of its publicly quoted international competitors.
The British-based company enjoyed the best year in its history in 2019 in revenue and profit terms.
Mr Crozier is leaving after just under three years as Vue's chairman.
He recently became chairman of Kantar, the market research business acquired by Bain Capital from WPP Group.
Vue is majority-owned by ?two Canadian funds: the Alberta Investment Management Corporation (AIMCo) and Omers, an Ontario-based pension fund manager.
They each own 37% of the company, with the remainder held by management and employees.
The business employs more than 5,000 people in the UK, and has said it will continue to pay them through the coronavirus crisis, despite the fact that it has not been able to secure rent holidays from landlords.
Vue trades from about 90 sites in Britain, and is valued by bankers at in excess of £2bn.
Like the rest of the industry, it is likely to face a bumpy financial ride given the uncertainty over when government freedom of movement restrictions and social distancing measures will be curtailed.
"Despite the extraordinary circumstances we find ourselves in as an industry, we are confident that we will survive, and are ready and excited to deal with the massive surge in customer demand once the current public health measures are lifted," Mr Richards said.
"Families, couples and individuals who have been cooped up at home for weeks or months will recognise more than ever that the big screen entertainment offers something they cannot find in their living rooms, and we are confident of an even brighter future for Vue and the movie industry in the near future."
A number of anticipated blockbuster films, including the latest James Bond outing - No Time To Die - have had their releases delayed until later in the year.
Meanwhile, streaming services such as Netflix have been forced to limit streaming quality to cope with a surge in demand across Europe.
Vue is the UK's third-biggest cinema operator by number of screens, trailing just behind Cineworld and Odeon.
Cineworld has seen its share price plunge in recent months, with the fall accelerating in the last fortnight as investors take fright at its increasingly strained balance sheet.