Labour is increasingly optimistic it can force concessions from the government over the status of EU nationals in post-Brexit Britain, after securing cross-party backing for an amendment to the article 50 bill.

Angela Smith, Labour’s leader in the Lords, said: “My sense is there’s real momentum behind this one, and my sense talking to peers across the house and listening to the debate is this is something that the House of Lords would support, cross-party.”

Lady Smith added that strong public disquiet about the future of EU citizens, and lingering concerns among Conservatives in the House of Commons about the issue, would help Labour’s cause.

Conservative peer Lord Bowness has added his signature to a clause tabled by Labour, calling for the government to guarantee the rights of EU citizens to remain in Britain.

The Liberal Democrats, who had drafted their own amendment aimed at securing the same result, have now thrown their weight behind Labour’s version, in the hope of creating enough cross-party pressure to force the government to move. The crossbench peer and former diplomat Lord Hannay is also backing the amendment.

Hopes of progress on the issue came as the Labour leader, Jeremy Corbyn, prepared to deliver a speech on how his party would fight to protect workers’ rights and prevent Britain becoming a “bargain basement” after Brexit.

Speaking at a conference of leftwing European politicians that was convened to discuss the implications of leaving the EU, Corbyn plans to come out fighting regardless of the results in a pair of key byelections in Copeland and Stoke-on-Trent Central.

He will tell his audience in London on Friday: “Theresa May’s government has wrapped Brexit up in fake patriotic posturing, waving the union jack while preparing to sell out our public services and consumer protections to US corporations, cheered on by the corporate media, who pile up their wealth in overseas tax havens while posing as national champions and fostering division both at home and abroad.”

Corbyn suffered a rebellion in the Commons when he whipped his MPs to back the article 50 bill.

But his allies believe it would have been electorally toxic to be seen to “block Brexit”, and now hope to turn the conversation to the kind of country Britain will be outside the EU.

Keeping up the pressure on the prime minister as the bill passes through the Lords is a part of that strategy, and the shadow Brexit minister, Keir Starmer, has worked closely with Smith and her team.

Theresa May has promised to tackle the issue of EU citizens’ rights early in the Brexit negotiations, but has insisted she wants to see the status of British citizens living elsewhere in the EU settled at the same time.

The issue was one of those raised as a concern by MPs, including Tories, as the legislation authorising the government to trigger the formal process for leaving the EU passed through the Commons.

Several Tory peers have expressed concern about the issue. Lady Altmann, a former government minister, told the Guardian: “I do understand the government’s position, and that it is hard negotiating when you have already given something away, but this is about people’s lives. People who have done nothing wrong, and who have contributed to this country.”

Lib Dem leader Tim Farron said: “The government know they are facing a defeat on this issue. Either they make concessions or this will return to the Commons.

“It is vital that EU citizens are given clarity about their future. The current position of the government is unfair and unsustainable. The government would do well to recognise that fact.”

Campaign group the3million lobbied parliament earlier this week about the fate of EU citizens already in the UK.

If the article 50 bill is amended by the Lords, it will bounce back to the Commons next month, in a process known as parliamentary “ping pong”. The committee stage of the bill will take place in the Lords next week.

If the Lords amends the legislation, the government would have to decide whether to cave in, or bounce the bill back to the upper chamber.

Smith said: “If they go down fighting on this, the signal that would send for the negotiations would not be positive.”

Labour’s amendment says the government must “resolve to ensure that citizens of another European Union or European Economic Area country, who are legally resident in the United Kingdom on the day on which this act is passed, are not disadvantaged in relation to their right to reside and work in the United Kingdom or their potential to acquire such rights in the future”.

Smith also said she hoped for a concession on whether the Commons would be offered a meaningful vote on the EU deal at the end of the government’s negotiations – another issue that exercised some Tory backbench MPs.

“Ministers have rightly conceded the need for a vote before the deal is concluded and we welcome that,” she said.

“Our amendment on a vote on the outcome of negotiations has been drafted by taking note of what happened in the Commons, including ministerial comments, as well as expert advice from peers in the Lords debates.

“Rather than ministers just holding to the line that they won’t amend the bill but will make a statement on the record, it would be a positive and mature signal for future progress of legislation if having accepted the principle, they write this into the bill.”

Downing Street refused to be drawn on how it would respond if it lost a vote in the Lords, where the government does not have a majority.

A spokesman for the prime minister said: “In terms of what the Lords do, clearly that’s a matter for the Lords. I can’t address something that hasn’t happened yet.”

He added: “On the broader issue of the rights of EU citizens, I think the PM has been very clear on this, that we want to guarantee those rights of EU citizens in this country, we want there to be reciprocal arrangements for UK citizens living in other member states.

“And we want this to be one of the things that is addressed early in the process.”