After a bumpy week of Brexit spats within her administration and with the EU, May wants to fend off another setback in a long-awaited showdown with restive lawmakers.
MPs in the lower House of Commons will vote on a raft of amendments produced by the upper House of Lords, which May claims would weaken the government's hand in exit talks with the EU.
"We must think about the message parliament will send to the European Union this week," May told MPs in her centre-right Conservative Party late Monday.
Her minority Conservative government relies on the support of Northern Ireland's Democratic Unionist Party for a slender working majority in the 650-member Commons.
"I am trying to negotiate the best deal for Britain. I am confident I can get a deal that allows us to strike our own trade deals while having a border with the EU which is as frictionless as possible," she said.
"But if the Lords amendments are allowed to stand, that negotiating position will be undermined."
Flashpoints
The EU (Withdrawal) Bill, coming back down to the Commons from the unelected Lords, is the draft law that would set the legal framework for Brexit.
The government wants the Commons to reject 14 out of 15 amendments introduced by the Lords which are intended to keep Britain close to the European Union after Brexit.
However, May is worried about the prospect of a rebellion by pro-EU Conservative MPs who are determined to retain as many of the changes as possible.
Flashpoints include proposals to increase the power of parliament to decide on the final Brexit deal, and others seeking to keep Britain tightly aligned with the EU's economy after it leaves the bloc.
Some europhile Conservatives were reported to be backing away, worried that if May was fatally damaged by defeat, it could open the way for a hardline Brexiteer to take over the party and the premiership.
May warned her MPs: "The message we send to the country through our votes this week is important.
"We must be clear that we are united as a party in our determination to deliver on the decision made by the British people."
Customs union concession
One amendment in danger of not being overturned on Tuesday is on the so-called meaningful vote, which would give parliament the power to decide what to do if it rejects the final Brexit deal.
On Wednesday, one on joining the European Economic Area -- the single market -- will likely fall because the main opposition Labour Party is against it.
However, the government may lose a vote Wednesday on membership of the EU's customs union, but this may not have much practical impact due to the way it is drafted.
Europhile Conservative Sarah Wollaston said she was "minded" to rebel.
"We would like to see further concessions on the amendment on the customs union because it is just a very sensible amendment that says keep it on the table, don't completely rule it out," she told the BBC.
After May addressed her MPs, Brexit minister Steve Baker indicated that the government would consider a new compromise amendment regarding customs relations.
"Our policy is to leave the customs union so that we can conduct our own independent trade policy but it would be appropriate that we have an arrangement in place with the European Union," he said.
The increasingly febrile atmosphere comes as pressure builds for a deal by October ahead of Britain's withdrawal from the EU in March next year.
There is a sense among both eurosceptics and pro-Europeans that crunch time is fast approaching.
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