Labour Lords sources have told Sky News they are 80-90% sure the Brexit bill will be passed by peers by midnight on Monday.
It will allow Theresa May to commence Brexit negotiations on Tuesday.
In the last few weeks, the Lords voted for two amendments to the Brexit bill: to give European citizens a guaranteed right to remain in the UK; and to give Parliament a vote on the withdrawal deal Theresa May negotiates with the EU.
The bill is going back to the Commons when it is expected to once again be approved without those amendments before returning to the Lords.
The source told Sky News that, this time, the Lords are almost certain to vote it through unamended.
It comes amid reports Cabinet ministers have been told to cancel overseas trips and Tory whips are making sure every MP is present for the signing of the bill in Parliament this week.
Theresa May has said she wants to invoke Article 50 by the end of March, with a target date of Wednesday 15 March, something she looks set to achieve.
Earlier, Jeremy Corbyn said he wanted Labour to continue to press for changes to the bill.
The Labour leader said the party should "keep on" fighting for the rights of European citizens.
Mr Corbyn urged the House of Lords to "stick with" their earlier stance.
Mr Corbyn said: "We will keep on with this. Clearly to win requires all opposition parties to unite on this, that we seem to be able to do. But it also requires Conservative abstentions or votes with us on it.
"I hope the Lords will stick with that position and send it back to the Commons, because it seems to be an important message to put that EU nationals make a massive contribution to our society... and they ought to be given the opportunity to remain in Britain.
"If the Commons does not accept those Lords amendments it goes back to the Lords. If they persist it comes back to us a day or two later. We will keep on with this."
His shadow chancellor John McDonnell said the Government should listen to the Lords and take on board what they have voted for.
He said: "We are hoping the Government will see sense. We can then get Article 50 agreed in a sensible way and get on with the real negotiations."
Sky's Political Correspondent Beth Rigby said: "It doesn't seem like there is the air of rebellion in the Lords. She could well be in a position to trigger Brexit on Tuesday."
European leaders are preparing for formal Brexit negotiations to begin within days should Mrs May trigger Article 50 next week.
EU leaders, meeting in Brussels, were told to prepare for the possibility that Britain could trigger talks as early as next Tuesday, with a gathering on 6 April pencilled in to respond to Britain's formal letter of notification.
Irish Prime Minister Enda Kenny confirmed to reporters that the EU 27 had provisionally agreed a meeting in early April to agree a framework for exit talks.