A total of 338 staff voted in the survey on Malcolm Gillies and his senior management.
The institution had 2,855 staff in the year 2010-11, according to the Higher Education Statistics Agency.
The chair of the governing board, Clive Jones, has said he stands “firmly behind” Professor Gillies.
The union said that the vote represented a rejection of what it claimed was Malcolm Gillies’ “unrelenting strategy to shrink, sell off or privatise the university’s physical, human and educational resource”.
Last year the university announced a radical programme of course closures, and in January it emerged that the institution would make more than 200 academics redundant to save ?7 million, which has prompted the threat of strikes from the UCU.
Professor Gillies has mooted sharing administrative services with other institutions to save money.
The UCU has also attacked Professor Gillies’ plans to ban alcohol from some parts of the campus because some Muslim students found drinking “immoral”.
“It is now clear that the current board of governors of London Met must intervene and re-direct the current flawed management of London Met,” the UCU said in a statement, adding that it was seeking a meeting with the board as soon as possible.
In response, a statement from the university said that the changes were driven by the “realities of increasingly constrained public funds and the special challenges of London Met’s own position” .
“London Met has a substantive job of work to do in delivering this plan, and it is doing it,” the statement says. “The recent survey by UCU and Unison, however they arrived at it, shows that this is a time of significant change”.
A spokeswoman for the university confirmed that Professor Gillies had asked all staff in a newsletter not to complete the survey because it was “unofficial”.