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.
The bill has already suffered two defeats in the House of Lords.
If MPs do reject the amendments, Parliament could sit through the night to try to reach an agreement.
However Mrs May has said she will take the UK out of the EU even if Parliament votes against the deal she is offered.
She has also said she wants Article 50 triggered by the end of March.
‘Done and dusted’
Both the House of Commons and House of Lords will debate the Brexit bill on Monday. MPs will debate and vote first, before it is passed to peers to agree or disagree with the decisions made.
Time has also been set aside for the debate on Tuesday and Wednesday, if necessary, for the MPs to consider the amendments.
The bill travels back and forth between the two chambers until both sides agree.
The bill will then go for Royal Assent, after which Mrs May can formally tell the rest of the EU that she is ready to start negotiating.
No deal preparations
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 with no deal in place.
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.
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”.
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.
Shadow Brexit secretary Sir Keir told Sky News that the prospect of the UK “crashing out without a deal” would be “a disaster”.
He urged the prime minister to consider keeping the “really important” Lords amendments – adding that EU citizens in the UK had been “left in limbo”.