The union wants Reuters to consult with union members to seek alternatives to a proposal for outsourcing and relocating jobs, which was announced earlier this month as part of the company's 'Fast Forward' programme.
Without a satisfactory response from Mr Glocer, union members will prepare a ballot for industrial action in early December.
Around 10 London-based staff at the Reuters website would lose their jobs, reducing the web team to around four people. A further 25 UK posts, including picture desk and graphics staff, could also be cut.
According to the UK's National Union of Journalists (NUJ), the Fast Forward programme will save Reuters £440 million per year - £340 million of which will be saved by cutting jobs. More job cuts are possible on the Reuters sports desk.
"The integrity of Reuters comes down to its sports and news - for years that has been the benchmark," said NUJ newspaper organiser Barry Fitzpatrick.
"We are flabbergasted that they think they can get away with this."
The NUJ is also in touch with union colleagues in India, where journalists may benefit from outsourced jobs.
"This is not only about outsourcing, it is about downgrading. They will be paid less, and won't have the same standards, no pensions that we know of, no health and safety regulation and no employment rights," said Mr Fitzpatrick.
A spokesperson for Reuters told dotJournalism that some reports of job cuts had been exaggerated, and that cuts on the UK site would involve staff responsible for formatting and packaging online content for the Reuters website.
Under the new proposal this work would be done remotely by an existing team in Toronto.
"Overall these changes will impact about 10 people," said a spokesperson for Reuters.
"We are still looking at budgets and our plans for next year, but we expect the head count to be minimal."
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