Ministers believe MPs will reject the two changes made to the Brexit bill in the House of Lords when they debate it for the second time later.
Peers want to guarantee the rights of EU citizens in the UK and ensure Parliament has a vote on any deal.
The EU Withdrawal Bill could complete its final stages if both Houses of Parliament agree the text of the bill.
PM Theresa May could then trigger Article 50, which formally starts the Brexit process, as early as Tuesday.
BBC chief political correspondent Vicki Young said she thought it likely that MPs would overturn the Lords’ amendments to the bill, and did not expect peers to try to block the bill any further.
This could mean it is all “done and dusted by midnight” on Monday, she said.
Both the House of Commons and House of Lords will debate and vote on the bill on Monday. MPs will go first, before it is passed to peers to agree or disagree with the decisions made.
The bill travels back and forth between the two chambers until both sides agree.
Labour has urged the prime minister to consider keeping the “really important” Lords amendments – adding that EU citizens in the UK had been “left in limbo”, waiting to hear if they would have the right to stay.
If MPs do reject the amendments, Parliament could sit through the night to try to reach an agreement, and time has also been set aside on Tuesday and Wednesday.
Once it is agreed, the bill will go for Royal Assent, after which Mrs May can formally tell the rest of the EU that she is ready to start negotiating.
In an interview on BBC One’s Andrew Marr Show on Sunday, Brexit Secretary David Davis insisted the UK would be prepared if it has to leave the EU at the end of the two-year negotiating process with no deal in place with the other 27 nations.
He appealed to MPs not to “tie the prime minister’s hands” over Parliament getting a final vote on the deal and on EU citizens’ rights in the UK.
Mrs May has said Parliament will get a vote, but some MPs and peers want a stronger commitment to give them a more “meaningful” say – and to be able to potentially send her back to the negotiating table in Brussels.
Conservative MP Nicky Morgan, a Remain campaigner, told BBC Radio 4’s Today programme seeking such an assurance was not about blocking Brexit.
“David Davis and everybody else know that Parliament will find a way to have a vote. Isn’t it better that the government acknowledges that today…?” she said.
Ms Morgan added: “If the prime minister wants a united party behind her this is a simple reassurance that can be given by ministers at the despatch box that will have the effect of me and my colleagues supporting the government on this.”
Foreign Secretary Boris Johnson told ITV’s Peston on Sunday that “no deal” would be “perfectly OK”, and the consequences were not “as apocalyptic as some people like to pretend”.
But Conservative former deputy PM Lord Heseltine, who was sacked as an adviser last week after rebelling, dismissed his comments as “rubbish”, saying many Conservatives felt “betrayed”.
Mrs May has said she would rather take the UK out of the EU with no deal rather than a “bad deal”.
No deal would result in trade rules defaulting to World Trade Organisation standards, meaning the UK and EU would be obliged to apply to each other the tariffs and other trade restrictions they apply to the rest of the world.
Analysis By Vicki Young, BBC chief political correspondent
Conservative MPs are looking for further verbal guarantees from ministers over the exact nature of the parliamentary vote they’ve been promised on any Brexit deal Theresa May negotiates.
Labour are telling their MPs to vote against the government and around 10 Tory backbenchers have defied orders from their party managers in previous votes on this bill, but a rebellion on that scale won’t be enough to defeat the government and ministers aren’t expected to make concessions.
All eyes will then move to the Lords where the Liberal Democrats have promised to keep the pressure on the government – but Labour peers seem more likely to back down, so by tonight the rebellion could have melted away.
Brexit bill: Ministers expect MPs to reject changes}