A law firm is taking action to ensure the formal process for the UK leaving the EU is not started without an act of Parliament.
Mishcon de Reya, lawyers acting for a group of business people and academics, said it would be unlawful for a prime minister to trigger Article 50 without a full debate and vote in Parliament.
It comes after the UK voted to leave the EU in the 23 June referendum.
Number 10 said Parliament should "have a role" in deciding the way forward.
Following the referendum, David Cameron announced he would stand down as prime minister by October and would leave his replacement to trigger Article 50 of the Lisbon Treaty.
Once the legal process is triggered there is a two-year time limit to negotiate an exit deal from the EU.
Mishcon de Reya's clients argue that under the UK constitution the decision to trigger Article 50 rests with Parliament.
The firm has been in correspondence with government officials to seek assurances over the process.
The BBC's legal correspondent Clive Coleman said Mishcon de Reya believed that any prime minister using executive powers to start the process would be acting unlawfully because they would be overriding the 1972 European Communities Act that enshrines UK membership of the EU.
The law firm says that constitutionally only legislation can override legislation and an act of Parliament is required to give the prime minister legal authority.
The passage of such an act could in theory provide the majority of MPs who favoured Remain the chance to block the UK leaving, our correspondent said, but he added that this seemed "constitutionally inconceivable".
Kasra Nouroozi, a Mishcon de Reya partner, said: "We must ensure that the government follows the correct process to have legal certainty and protect the UK Constitution and the sovereignty of Parliament in these unprecedented circumstances.
"The result of the referendum is not in doubt, but we need a process that follows UK law to enact it.
"The outcome of the referendum itself is not legally binding and for the current or future prime minister to invoke Article 50 without the approval of Parliament is unlawful.
"We must make sure this is done properly for the benefit of all UK citizens. Article 50 simply cannot be invoked without a full debate and vote in Parliament."
By Clive Coleman, BBC legal correspondent
It has come as a shock to many that the referendum result itself is not legally binding in UK law and it alone does not trigger the UK's departure from the EU.
That has to be done under the withdrawal process laid down in Article 50 of the Lisbon Treaty.
It's argued that a prime minister acting alone under prerogative powers lacks the constitutional authority to trigger Article 50 and an act of Parliament would need to be passed giving him or her that authority.
The passage of that act would of course provide the opportunity for MPs (a majority of whom favour Remain) to express their views on Brexit and in theory vote according to their consciences.
However, it seems constitutionally inconceivable that Parliament would fly in the face of the Leave vote secured through a national referendum and refuse to pass an act that gave the prime minister authority to begin the "divorce" process.
In other words, the referendum has changed nothing legally but everything politically.
European leaders have said the UK should not delay leaving the EU, with European Commission President Jean-Claude Juncker saying the UK does not have "months to meditate" on activating Article 50,
The two-year negotiation period under Article 50 can be extended only with the unanimous agreement of the remaining 27 member states.
If there is no extension, the UK ceases to be a member of the EU on the conclusion of an agreement within the two years, but in any event two years after notice has been given.
David Cameron's spokeswoman said triggering Article 50 was "a matter for the next prime minister".
A Cabinet Office spokesman said: "As the prime minister said in the Commons, we have now got to look at all the detailed arrangements, and Parliament will clearly have a role in making sure that we find the best way forward.
"It will be important to ensure in moving ahead that the interests of all parts of the United Kingdom are protected and advanced."
Ex-defence secretary Liam Fox, who is bidding to become the next Tory leader, said any MP that voted against the referendum result "does not deserve to have a place in the House of Commons".