Ministry of Justice spokesman Lord Keen of Elie said, while many wanted to remain citizens of the EU, more had decided they did not in the Brexit referendum.
He told peers: ”It’s simply not feasible for us to set upon a course of negotiation that is doomed to failure.
“We cannot secure EU citizenship for citizens of the UK after we leave the EU.”
Lord Keen was responding to a call by Plaid Cymru’s Lord Wigley for ministers in Brexit negotiations to secure continued EU citizenship for UK citizens.
Warning there was strong opposition to its loss, Lord Wigley said associate European citizenship was a model the Government could adopt.
He claimed: ”This is all a matter of political will.
But Labour’s spokeswoman Baroness Hayter of Kentish Town said that while many “ached” to retain EU citizenship, “standalone EU citizenship” did not exist.
Lady Hayter said sadly UK citizens would lose EU citizenship under Brexit but added: “What we can do is to ensure that in every moment of the negotiation we prioritise the movement of people around the continent in a way that a generation has learned to enjoy and value.”
She added: “It is a continental Brexit that we would like to see.”
The exchanges came as weary peers neared the end of 11 days of detailed line-by-line study of the Bill, which transfers EU law into UK domestic law ahead of exit day.
During the debate, it was claimed peers will have done a worse job than their 16th century predecessors if they allow so-called Henry VIII powers to pass through in the legislation.
Liberal Democrat peer Lord Tyler joined former Commons clerk Lord Lisvane in sounding the alarm over powers granted to ministers in the flagship Bill.
Lord Tyler, a former Lib Dem MP, said even Henry VIII’s proclamation powers were repealed in 1547 after his death.
He said: ”As I understand it, our 16th century predecessors actually did put around the statute of limitations some additional restrictions which are not here in this Bill.
“As a former Member of the other House, I really think it is extraordinary that the two Houses of Parliament, if we allow this to go through, it may seem a minor matter, but in terms of a precedent, I think it is extremely important.
“We do not seem to have done our job, if we let it through, as well as our 16th century predecessors.”
The latest row focused on a technical aspect of what EU laws will be reproduced in official documents after Brexit.
Under the Bill, EU laws will not have to be published if a minister is satisfied that a relevant instrument has not become or will not become on exit day retained direct EU legislation.
The Bill allows this to be done by ministerial direction.
Lord Lisvane cited criticism from the Lords Delegated Powers and Regulatory Reform committee, which said amending the law by direction was highly unusual.
But Baroness Goldie told peers that introducing exemptions from publishing certain instruments was a “targeted, common sense provision”.
A “targeted, carefully circumscribed power” for ministers to give such direction was neither unprecedented or harmful, she added.
She said: ”What this is all about is trying to ensure that the statute book does not become cluttered with material which is irrelevant, not competent on the Bill and not within the scope of retained EU law as we have defined it.
“There has to be common sense applied. There has to be some proportionate way of balancing the legitimate interest in the constitution with the practical need to make sure that we don’t create nonsense on the statue book.”