Sources said that the presenters of the hit BBC Two series had not yet signed their new contracts even though their current agreements expire at the end of the month.
It has been claimed that there has been "a meltdown" in negotiations and that the pair "turned down flat" the first offer from the BBC.
They were said to be arguing that they deserve a pay package similar to that enjoyed by Jeremy Clarkson, their co-presenter.
Mr Clarkson is thought to have struck a multi-million pound deal last month with BBC Worldwide, the commercial arm of the corporation, which on top of his wages would give him a stake in the profits made from exploiting the Top Gear brand in the UK and abroad.
His new salary is said to put him among the highest-paid presenters at the BBC, alongside Jonathan Ross, who reputedly earns £18 million over three years, and Graham Norton, who is thought to take home £5 million over two years.
A source said yesterday: "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.
"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."
He added: "They don't want to accept second status anymore. There has been a meltdown in negotiations."
The negotiations come at a difficult time for the BBC, which is having to make cuts in key departments to make up for a funding gap left by a less-than-expected licence fee settlement.
The BBC Trust, the corporation's governing body, published a review last month into the pay of the BBC's top talent, which concluded that it was not paying over the odds for on-screen and on-air presenters.
However, Sir Michael Lyons, the trust's chairman, said that the BBC had to be prepared to "walk away from deals that do not offer value for money"
Mr May laughed off suggestions that he had not signed his new contract because he was angry about Mr Clarkson's new salary.
"The only reason I haven't signed my contract yet is because it has not yet been finalised," he told The Daily Telegraph. "The current one doesn't expire until the end of July. I'm sure a new one will be sorted out soon."
He added: "I'm not stomping around about Jeremy's contract with BBC Worldwide. Quite frankly, I couldn't give a **** about it."
A spokesman for Mr Hammond declined to comment.
A BBC spokesman said: "We never comment on contract negotiations."
