North Westminster politicians are campaigning to save the building, which survived the construction of the nearby Westway, which ploughed through the area in the late 1960s.
A 1,100-signature petition against the proposal was handed to Transport for London last month.
Labour Party London Assembly Member Murad Qureshi said the announcement that the shop would be demolished to make way for an electricity sub-station being built as part of the project had come out of the blue.
He said: “Essentially I have been a supporter of Crossrail but had not expected it to cost local jobs and a local landmark like the Big Table.
“Crossrail have had more then 20 years to sort these issues out and should not have had to sort them out only a few months before the tunnelling begins.”
At a meeting last week, residents said they were concerned about a lack of public consultation and what they called the “failure” of Crossrail to explain its plans.
A Crossrail spokesperson said: “We are sensitive to the concerns of the Big Table. Crossrail is making every effort to enable the business to stay at its leased premises for as long as possible.
“We have recently advised Big Table that they will be able to stay in their existing premises until 2013.”
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