The first is a "joint legally-biding instrument" on the withdrawal agreement.
Prime Minister Theresa May's government has been seeking changes, but the European Union refuses to reopen the agreement after long negotiations.
He appealed for unity among MPs and the country, and rejected the notion of a no-deal Brexit: "We didn't vote to leave without a deal".
Theresa May has arrived in Strasbourg for last-minute Brexit talks with European Commission president Jean-Claude Juncker on the eve of a crucial vote in the House of Commons.
The House of Commons overwhelmingly rejected the deal in January, primarily because of concerns over arrangements for the Irish border.
The campaign group, fronted by former Ukip leader Nigel Farage, has argued that not taking part in the elections could mean that the United Kingdom no longer has representatives in the European Union whilst Brexit negotiations are ongoing.
It comes as the government is making a last-ditch bid to strike a deal with Brussels on the controversial Northern Ireland backstop, with Attorney General Geoffrey Cox set to update legal advice on the agreement in a bid to win over the DUP and Tory Brexiteers.
Mrs May's efforts to secure new assurances appeared to have hit stalemate on Monday morning, as Mr Barnier said discussions were now taking place "between the Government in London and the Parliament in London", rather than the United Kingdom and Brussels.
The shadow Brexit secretary, Keir Starmer, immediately cast doubt on whether the changes would actually be legally binding, saying: "It sounds again that nothing has changed".
Silva charged with improper conduct after confronting referee
We have to keep the momentum but obviously, it's still not done. After we conceded we gave some belief to them. Then the third goal was a clear offside.
The prime minister was warned that her position could be in jeopardy unless she stuck to commitments she made about this week's votes.
Senior Labour backbencher Yvette Cooper said the PM would be guilty of a "straight-up lie" if she failed to go through with votes allowing MPs to delay Brexit.
Some called on the PM to postpone the "meaningful vote" rather than risk another damaging reverse.
The prime minister has secured "legally binding" changes to her Brexit deal a day ahead of MPs voting on it, says the Cabinet Office minister.
They urged her to table a "conditional" motion setting out the terms for dealing with the backstop issue which parliament would be prepared to accept.
"As it stands her deal is going to be defeated", a party source told the newspaper.
Former chief whip Andrew Mitchell told the Times of London that "anything that avoids what looks like a massive defeat on Tuesday is worth considering".
Such a disorderly no-deal Brexit would be a disaster for the British economy and would also hurt the EU, Tajani said, adding that he would be happy if Britain were to remain in the bloc.