Ministers will say it creates a unique partnership with the EU, while Labour argues it will make people poorer.
Late on Wednesday, chief whip Julian Smith tried to win over pro-Brexit Tories, while civil servants will brief senior MPs on a no-deal scenario later.
However, one Brexiteer complained this was a bid to "spook grandees".
Before the third of five days' debate, it has been suggested by some Conservatives that the prime minister ought to postpone next Tuesday's vote to avoid defeat.
A government source told the BBC that whips are looking at all options to deliver a majority.
BBC political correspondent Nick Eardley said there was "fervent speculation around Westminster" about what ministers could offer to win support, including giving backbench MPs more of a say on whether the controversial "backstop" is adopted or not.
However, he added: "As things stand delivering parliamentary approval for the Brexit deal still appears a tall order."
The Daily Telegraph reported that the EU could be prepared to discuss extending Article 50 - delaying Brexit until after 29 March - if the deal was rejected by MPs.
Meanwhile, Northern Ireland's Democratic Unionist Party, which props up Mrs May's administration, has said it would support the government in a confidence motion if the deal was thrown out.
The backstop is designed to protect the Northern Ireland peace process by preventing the return of customs posts and checkpoints at the Irish border, in the event a future UK-EU trade deal was not agreed.
However, while it would keep the entire UK temporarily under EU customs rules, it would require some new checks on goods transported to Northern Ireland from Great Britain, which the DUP says is unacceptable.
The full legal advice presented to cabinet before the Brexit deal was agreed was published on Wednesday, after the government lost a bid to keep it confidential.
It revealed the chief law officer's opinion that the backstop risked a "protracted rounds of negotiations" with the EU, could potentially last "indefinitely", and that the UK could not "lawfully exit" without EU agreement.
The DUP said this would be "devastating" for the UK.
The chairman of the pro-Brexit Conservative backbench European Research Group (ERG), Jacob Rees-Mogg met DUP Westminster leader Nigel Dodds on Wednesday.
The Press Association quoted Mr Rees-Mogg telling the ERG later that if Mrs May's deal went through, the Conservatives risked losing the DUP's backing, raising the possibility of a general election being triggered.
Thursday's debate will be opened at 11:30 GMT by Chancellor Philip Hammond, with International Trade Secretary Liam Fox expected to round off proceedings after about eight hours.
On Wednesday, Mr Hammond told the Commons Treasury committee: "When there is a deal on the table that has very, very modest costs to the economy, which will allow us to move on as a nation both economically and politically, I judge that even narrowly, economically that will be in the best interests of the country."
Mr Fox has warned of a risk of Parliament trying to "steal Brexit from the British people", after MPs voted to give themselves a greater say over the process.
He said there was a "natural Remain majority" in Parliament and that any attempts to delay the UK's departure or overturn the 2016 referendum result would be a "democratic affront".