Ms Reeves, chairwoman of the Business, Energy and Industrial Strategy Committee, has warned the choice between Theresa May’s Chequers plan and a no-deal Brexit was a “false choice.”
She said jobs and investment “will flow out of this country” if Britain does not remain in the single market and customs union.
Her comments came as Mrs May potentially faces a knife-edge vote in the Commons on any Brexit deal she finalises with the European Union in the coming weeks.
Speaking to the Press Association, Ms Reeves said: “Well, I hope that those in the Conservative Party like Anna Soubry stick to their guns this time because there have been too many occasions in the past where... they’ve rolled over at the last minute. I think it was very unfortunate that Dominic Grieve didn’t in the end back his own amendment because it would make it easier to amend the Brexit deal if he had’ve stuck to that, so the procedures for getting a better deal or rejecting the Government’s deal are going to be harder than they would have been if Dominic Grieve had’ve stuck to his guns.
“And that’s the worry I guess really, that up until now the ERG (European Research Group) has never backed down, but those with more Remain sympathies have.
“There’s actually more people with those sympathies, with the more Remain sympathies in Parliament or soft Brexit sympathies, than there are for the hard Brexiteers, it’s just the hard Brexiteers are more ruthless.”
Red lines set out by Mrs May at the beginning following the triggering of Article 50, she said, had “totally hampered the process” and made it “a lot harder to reach a compromise”.
Ms Reeves said: “I think the Prime Minister really wants to get a deal.
“She doesn’t want to be the Prime Minister who’s led us off a cliff and I think she also thinks it’s her duty to do that.”
She added: “I won’t vote for a Chequers type agreement... I think it is a false choice to say the choice is between Chequers and no deal.
“I hope the PM can do a deal with her European partners, but unless it meets some of the red lines that my party has set out and that I’ve spoken about, I won’t be able to support it.
“Because unless we remain in the single market and customs union, I think that jobs and investment will flow out of this country and I can’t with good conscience vote for a deal that I know will make my constituents and the country worse off.”
Ms Reeves said there were other alternatives, including extending the negotiating period or going back to the country, adding: “I think that what’s in front of us at the moment in terms of Chequers or a no deal are both outcomes that most people in the country would reject and so I think that people should be given a chance to have their say on a Brexit deal which will change the direction of this country for at least a generation to come.”
She also voiced concerns over how the EU referendum was fought.
Ms Reeves said: “We have strict rules in this country, and rightly so, around campaign finance and if they weren’t adhered to and if the referendum was won by foul means, that does call into question whether the referendum itself was legitimate.”
A no-deal Brexit, Ms Reeves warned, would be an “absolute disaster”, adding that the BEIS committee took evidence from manufacturing businesses who voiced “huge concerns” about customs checks and decoupling from the EU regulatory framework, particularly in the automotive sector.
She said businesses were “very reluctant” to talk about the risks of Brexit for “understandable reasons”, with some not wanting to “sour” relationships with Government, others not wanting to send a signal to shareholders that there were big risks ahead, plus considerations over not wanting to create uncertainty for their workforce.
The Labour MP said: “They are not scaremongering, they only speak out when they think that there is an existential risk to the way in which they run their businesses and I wish that businesses would speak out more and would have spoken out earlier.
“My experience of business is they only speak out on political issues when they really think there is no alternative to doing so.”
On Labour’s stance, Ms Reeves said she was “one of the people that’s been pushing our party leadership into a more pro-People’s Vote position”, adding Labour’s EU referendum campaign “could have been stronger and I think it could have been better led from the top of the organisation.”