Liberal Democrat, some Labour and SDLP MPs have told the BBC they are prepared to vote against triggering Article 50.
Lib Dem leader Tim Farron said his party would oppose it, unless they were promised a second referendum on the UK's Brexit deal with EU leaders.
Several Labour MPs are also willing to vote against it, despite the Labour Party pledging not to do so.
The government says Lib Dem and Labour MPs are "trying to thwart and reverse the referendum result".
With the support of Conservative MPs and the support or abstention of most Labour MPs, the bill is well placed to pass through the Commons.
But the opposition of some MPs is likely to embolden critics in the House of Lords.
The Liberal Democrats have long called for a referendum on the outcome of the government's negotiations with EU, but only now have they said they will definitely vote against Article 50 if their demand is not met.
Mr Farron, whose party has eight MPs in the Commons, told BBC Radio 4's Today: "Article 50 would proceed but only if there is a referendum on the terms of the deal and if the British people are not respected then, yes, that is a red line and we would vote against the government."
For Labour, shadow minister Catherine West, former leadership contender Owen Smith and south London MP Helen Hayes all made clear they were prepared to vote against Article 50 - which begins formal exit negotiations with the EU - if amendments were not accepted.
Former Labour minister David Lammy and shadow transport minister Daniel Zeichner have said they would oppose Article 50. Opposition whip Thangam Debbonaire said she would also vote against it, if a vote were held imminently.
The SNP's 54 MPs may join them. First Minister Nicola Sturgeon has said they will not vote for anything that undermines the will of the Scottish people, and has previously said they will vote against a bill to write EU provisions into British law to prepare for Brexit.
Dulwich Labour MP Hayes said she was prepared to defy Labour whips to oppose the measure unless the government promised a second referendum.
She said: "I had somebody in my surgery last week who was in tears because of Brexit and I see genuine distress amongst my constituents about what this path means.
"I would not be representing them if I voted to trigger Article 50 on the basis of no information from the government about the path that they would then take us on."
In posts on Twitter and Facebook earlier this week, shadow Foreign Office minister Catherine West wrote: "As I have said before, I stand with the people of Hornsey & Wood Green, and I will vote against Brexit in Parliament."
Owen Smith confirmed to Today that if his bid for a second referendum failed, he was likely to oppose the bill.
The SDLP's three MPs will also oppose the measure.
Ministers said MPs voting against Article 50 would effectively be trying to re-run the referendum in the hope of a "different answer".
"Parliament voted by a margin of six to one to put the decision on whether to remain in or leave the EU in the hands of the British people," said Brexit minister David Jones.
"Only the Conservatives can be trusted to respect the outcome of the referendum and make a success of Brexit."
Last week the High Court ruled Parliament must be consulted about leaving the European Union.
Unless the Supreme Court overturns the judgement in December, a bill to invoke Article 50 is expected in the new year.
Labour made clear its official position would be not to frustrate the process of leaving the EU after a newspaper report said the party leader Jeremy Corbyn intended to force a general election unless ministers caved in to demands.
After the story broke Labour sources said that while it would seek to amend the bill, it would provide "unconditional" support.
Shadow Brexit secretary Keir Starmer said Labour would not frustrate the process and would not vote down Article 50.
However, Labour and Liberal Democrat peers will try to amend the bill in the House of Lords. So too will one Conservative peer - Baroness Wheatcroft.