The ex-prime minister has been speaking to his successor about what may happen if her proposed deal is vetoed by MPs.
Several ministers want MPs to be allowed to express their views on “Plan B” options to try to find a consensus.
Downing Street said there were “no plans” for this and Mrs May was focused on getting MPs’ support for her deal.
But the BBC understands Mr Cameron has been speaking to the prime minister about how a series of “indicative votes” on various different Brexit outcomes may be handled if her deal is rejected.
In a rare public comment on the Brexit process last week, Mr Cameron, who campaigned to remain in the EU in the 2016 referendum and quit after the public voted to leave, said he supported the prime minister but was “worried” by signs that Parliament was not willing to approve her deal.
Last week Mrs May called off a Commons vote on her Brexit deal, admitting it was likely to be heavily rejected.
The UK is due to leave the EU on 29 March 2019 – the deal sets out the terms of exit and includes a declaration on the outline of the future relations between the UK and the EU.
But it only comes into force if the UK and European Parliaments approve it.
In a statement later to MPs updating them on Thursday’s talks with European leaders, Mrs May will say holding another referendum on Europe, as some have suggested, would “break faith with the British people”.
The prime minister has signalled MPs will now vote on the deal on the table early next year, and no later than 21 January.
But Labour and other opposition parties, as well as some Tory Brexiteers, say a decision is needed now, so alternative options can be considered if Mrs May’s deal is rejected.
They are seeking to force a vote before the Christmas recess begins on Thursday, although the BBC’s Norman Smith said it was not clear how they could do this.
Potential “Plan B” options include:
- pursuing different Norway or Canada-style arrangements with the EU;
- leaving on the basis of a “managed no deal”;
- delaying Brexit to restart negotiations;
- hold a fresh referendum.
The PM is coming under pressure from cabinet ministers to “test the will of Parliament” through a series of “indicative” non-binding votes – which would see MPs pass judgement on the options available in the hope of identifying the most popular and shaping the way forward.
The most recent examples of this happening were a series of votes on proposed reforms to the House of Lords in 2003 and 2007 – when all the propositions were rejected.
The prime minister is due to update MPs at 15:30 GMT on last week’s European Council summit, where she made an appeal to EU leaders to help get the deal “over the line”.
The PM, who survived a vote of no confidence in her leadership by Conservative MPs last week, is seeking legal assurances over the controversial “backstop” plan to prevent a hard border between Northern Ireland and the Irish Republic, if no trade deal is reached.
EU leaders told her the negotiated withdrawal agreement was “not open for renegotiation”, although some clarification was possible.
Mrs May has insisted “further clarification and discussion” was possible and that the UK would be “working expeditiously over the coming days to seek those further assurances I believe MPs will need”.
But Labour leader Jeremy Corbyn said her deal was now “dead in the water” and the prime minister had “utterly failed in her attempts to deliver any meaningful changes”.