If the PM’s Brexit deal cannot get MPs’ backing Parliament “will have to decide on the alternatives”, Liam Fox said.
The cabinet Brexiteer said if there was no change on the backstop issue “it’s unlikely to pass through Parliament”.
He said another referendum was unlikely but did not rule out MPs getting a “free vote” on different ways forward.
Theresa May has accused the former Labour PM Tony Blair of undermining Brexit negotiations by calling for another referendum.
She met EU leaders on Thursday – after postponing a Commons vote on the withdrawal deal she has negotiated with the EU, fearing its heavy defeat.
The government says the Commons vote will go ahead in January, as talks continue with the EU on the issue of the Irish border “backstop”.
The backstop is an “insurance policy” in the withdrawal deal to prevent the return of a hard border with Northern Ireland if no trade deal is reached – but many of Mrs May’s MPs say they cannot support it.
EU leaders have said the deal is “not open for renegotiation” – but that there could be some further clarification.
International Trade Secretary Mr Fox told BBC One’s Andrew Marr Show that talks would continue with Brussels over Christmas and the New Year about how the backstop could be operated “in a way that is acceptable to both sides”.
“Clearly if there is no reflection of the anxieties that MPs have had, of potentially being locked in to the backstop without any choice, it’s unlikely to pass through Parliament.”
Mr Fox said it was “clear” that the EU understood the problem, and it was now a question of finding a “mechanism” that would remove those concerns.
If that mechanism is not found, he said: “The question is, would it be worth putting something to the House of Commons, knowing it would be rejected?”
If the deal could not get through the Commons, he said: “Parliament would have to decide on the alternatives.”
Other options backed by different groups of MPs include leaving without a deal, another referendum, or Norway or Canada-style alternative deals.
Mr Fox said there were problems with another referendum – including that it would “perpetuate division in the country” and that Parliament had pledged to “honour” the result of the 2016 referendum.
He added: “Supposing we had another referendum. Supposing the Remain side won it by 52-48 but it was on a lower turnout – entirely possible… people like me will be immediately demanding that it’s best of three – where does that end up?”
He claimed there were Labour MPs who backed the government’s deal but were being ordered to vote against it.
Asked about the potential for MPs to get a “free vote” – where MPs are not directed by their parties – he said: “That’s not something we have considered. I have to say, personally, I wouldn’t have a huge problem with Parliament as a whole having a say on what the options were.”
Meanwhile, Labour frontbencher Andrew Gwynne told the BBC the party would be using “parliamentary tactics” to try to bring the MPs’ “meaningful vote” on the deal forward to this week.
“We can’t move to the next stage until Parliament has decided whether or not to back the prime minister’s deal,” he said.
Asked whether his party would campaign for Brexit under a Labour deal if there were to be another referendum on the issue, he said: “Let’s wait and see. These things are moving very quickly.
“We are a democratic party and we will put our decision to the party members in a democratic way before we decide what the next steps are.”
Source BBC News