Logistics firm DHL has been holding talks with the staff union, which is now expected to give notice of industrial action on Tuesday.
According to The Sunday Times, DHL contacted JLR chief executive Ralf Speth on Friday to inform him that it had failed to reach an agreement with Unite, the union.
DHL, which employs 1,800 people at JLR's three main plants in the country, manages warehouses and the delivery of parts to production lines.
A logistics blockade could bring operations at Solihull and Castle Bromwich in the Midlands and Halewood on Merseyside to a standstill, forcing a shutdown.
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The first vehicle to roll off the Chinese production line will be the popular Evoque "baby Range Rover", currently made at Halewood.
"The management of Jaguar Land Rover can't make any assurances while this DHL issue is going on," the newspaper quoted a source as saying.
The Tata Group firm generates export revenues of almost 11 billion pounds a year and is one of Britain's largest employers, with 24,000 workers.
The company is also in advanced stages of plans to increase the size of its Halewood plant, which could result in creating up to 1,000 new jobs.
DHL employees, who often work alongside better-paid JLR workers, want a large pay rise to put them on similar terms.
DHL has reportedly offered a 4.5 per cent increase starting from January this year and 3 per cent or the rate of inflation in 2014.
However, Unite wants a 12.8 per cent rise over two years for workers who bring parts to the line, plus 20.6 per cent for drivers over the same period.
In 2008, the Jaguar Land Rover company was established when Tata Motors acquired the Jaguar and Land Rover businesses from Ford.