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The pair, whose contracts expire at the end of the month, are said to want pay deals more in line with co-host Jeremy Clarkson, which the BBC may not be prepared to meet.
"Internally there is talk that there might be a chance for the first time that there could be a break up of the team, it has got that bad," MediaGuardian quotes a source as saying.
"There has been a lot of pulling out of hair, a Mexican stand-off. For the first time it is possible that one or both might leave."
The source added: "They don't want to accept second status any more. There has been a meltdown in negotiations."
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