Peers should "not throw in the towel early" over Brexit, Peter Mandelson has said as he warned ministers they face defeats over key negotiating principles.
The Labour ex-cabinet minister believes the Lords will force changes to legislation intended to allow Prime Minister Theresa May to have the power to start the formal two-year Brexit process.
He insisted there is a "strong body of opinion" among peers over guaranteeing the future of EU nationals living in the UK and in giving Parliament a vote on the final Brexit deal in which the Government could be sent back to negotiate a better arrangement.
Ex-EU commissioner Lord Mandelson added the House of Commons "must prevail" on the European Union (Notification of Withdrawal) Bill as he also cautioned against a "Brexit at all costs".
His comments came after Conservative former minister Dominic Raab urged peers with generous EU pension pots to ensure they "came clean" and declared these when speaking in Parliament on Brexit.
Two days of debate on the Bill in the Lords will occur on Monday and Tuesday before amendments are considered the following week.
The Government lacks a majority in the Lords, with 252 Conservatives among the 805 peers - giving the opposition and independent crossbenchers a chance to inflict defeats.
MPs approved the draft legislation unamended and with an overwhelming majority earlier this month as Mrs May seeks to start Brexit talks before the end of March.
Asked if his view was that the Government could be defeated over the issues of EU nationals and how Parliament votes on the final deal, Lord Mandelson told BBC One's Andrew Marr Show: "I think it is.
"I think there's a strong body of opinion, across party and amongst the independent peers as well, that both these issues are very serious.
"But of course when it comes to EU citizens, the British Government is not negotiating with itself and there will be people among the member states who say 'No, we don't want to take this issue now, we'll take it later on during the course of the negotiations' - because it's as much a negotiating gambit for them as it is for Britain."
Questioned if there will be a long parliamentary battle between the Commons and Lords over the Bill, Lord Mandelson said: "At the end of the day, the House of Commons must prevail because it is the elected chamber.
"But I hope the House of Lords will not throw in the towel early."
Pro-EU Lord Mandelson, referring to the outcome of the Brexit talks, said: "We're going to have less trade, we're going to be paying through the nose for it and, broadly speaking, we're going to have the same number of immigrants coming to the country as well.
"Is that a reasonable deal? Don't you think the public will have something to say about that outcome when the negotiation ends? I think they will."
He also claimed: "There is no trade agreement in the world that will give us the same benefits in trade that we have now if we were to follow the Government's Brexit at all costs negotiation."
Justice Secretary Liz Truss said Lord Mandelson was a "blast from the past" and needed to move on from the referendum result.
She added that peers should "recognise the will of the people" and the "overwhelming" vote for the legislation in the House of Commons.
She told the same programme: "I just listened to Peter Mandelson, it was like the referendum had never happened.
"He seemed to be making the same argument that was being made last year and which the British people rejected.
"He needs to move on, and the Labour Party need to move on, because we are now in a new reality. He is a blast from the past, frankly."her positions at this stage."