David Davis will try to force an Article 50 Bill through Parliament
High Court judges ruled in November Theresa May has to consult Parliament and the devolved powers of Scotland and Northern Ireland before officially triggering Article 50, a decision her government has challenged.
Number 10 is expecting judges will find in favour of the original ruling, and have drawn up a short Bill in anticipation.
Mr Davis will head to the Commons shortly after the verdict on Tuesday if, as expected, it goes against the Government's wishes.
He will outline details of an emergency bill to be fast-tracked through the Commons and the Lords in order for Prime Minister to stick to her Brexit timetable.
Mrs May has previously pledged to trigger Article 50 by the end of March.
She had hoped the result of the EU referendum would give her the power to trigger Article 50 using a royal prerogative.
But those plans were scuppered when Remain campaigners brought a successful legal challenge to the High Court, arguing the process could not begin without Parliamentary approval.
Theresa May wants to trigger Article 50 by March
Although Labour leader Jeremy Corbyn has urged his MPs not to vote against Brexit, several rebel backbenchers have said they are considering it.
Mr Corbyn's former leadership rival Yvette Cooper backed his calls, saying any MP who defies the referendum result would be behaving like Donald Trump – who repeatedly said he would not respect the result of the US election if he lost.
Labour peer Peter Hain also told the Guardian he and a goup of colleagues will vote against triggering Article 50 when it reaches the House of Lords, saying a hard Brexit will make Britain "poorer for the next 10 years".
And in November, following her shock by-election victory over incumbent Zac Goldsmith, Richmond Park's Liberal Democrat MP Sarah Olney said she would vote against an Article 50 Bill because her constituency voted overwhelmingly to Remain.
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The Supreme Court will make a decision on Tuesday
Until last week, there had been concerns that many anti-Brexit MPs would use the Parliamentary Bill as a way to disrupt and delay the process of withdrawal from the EU until Mrs May shed more light on her Brexit plans.
Mrs May appears to have allayed some of those worries when she laid out her 12 point plan which included leaving the single market and the customs union in favour of control over immigration.
However, Mrs May still faces a backlash in her own party, with pro-Remain Tories reportedly mobilising to challenge her plans.
Former health secretary Stephen Dorrell, now chairman of pro-EU European Movement, described Brexit as a "pointless exercise" in an email to supporters and urged them to help the organisation "reverse course".
Ken Clark and Anna Soubry also challenged Mrs May's plans in the Commons, arguing for MPs to have a greater say on how Brexit is negotiated.