The Brotherhood is an Islamist movement which has been declared a terrorist group by Egypt's military rulers.
Reports suggested members of the group moved to London to escape a crackdown in Cairo and plotted a response to the ousting of President Mohammed Morsi, a Brotherhood leader.
There has been no reaction yet from No 10 regarding the lawyers' announcement.
The organisation's backing helped install Egypt's first civilian president, Mr Morsi, in 2012, but he was ousted in a military coup last year after widespread street protests.
The Muslim Brotherhood was founded in Egypt but now operates in many states and has influenced other Islamist movements around the world with its model of political activism combined with Islamic charity work.
The prime ministerial review ordered by David Cameron back in April into the activities of the Muslim Brotherhood in the UK was always going to be controversial.
A number of Arab countries, notably the UAE, Saudi Arabia and Egypt, all see the Brotherhood as a threat to their rule and want the UK as their ally to ban it.
There have even been veiled hints from the Gulf that trade deals with the UK could suffer if the review did not result in a ban.
The report's author, Sir John Jenkins, UK ambassador to Riyadh, spent three months taking testimony from a number of sources, including secret intelligence, before delivering his findings to No 10 where they have remained unpublished since July.
Whitehall insiders have been saying privately that - while there are concerns about some individual members - nothing has emerged to link the Brotherhood as an institution to any acts of terrorism.
While the Brotherhood - or al-Ikhwan al-Muslimun in Arabic - says it supports democratic principles, one of its stated aims is to create a state ruled by Islamic law, or Sharia.
Its most famous slogan, used worldwide, is: "Islam is the solution."
In December 2013, the new Egyptian government declared the Muslim Brotherhood a terrorist group after blaming it for an attack on a police station that killed 16 people.
Mr Cameron commissioned the review into the group's UK links in April.
The review was led by Sir John Jenkins, the UK's ambassador to Saudi Arabia - another country which has declared the Brotherhood a terrorist organisation.
At the time, Downing Street released a statement which said: "The prime minister has commissioned an internal government review into the philosophy and activities of the Muslim Brotherhood and the government's policy towards the organisation."
Since Mr Morsi's overthrow, more than 1,000 people have been killed and thousands of Muslim Brotherhood supporters detained in a crackdown by the interim authorities which has been condemned by human rights campaigners.

Wishaw General in North Lanarkshire experienced the power cut between 08:45 and 16:00.
It meant one emergency patient was redirected to Hairmyres Hospital. NHS Lanarkshire said IT systems were also affected.
The hospital was powered by back-up generators until mains power was restored.
NHS Lanarkshire chief executive Calum Campbell said: "In the interest of patient safety and as a precautionary measure, we took the decision to reschedule a small number of surgical procedures based on clinical priority.
"The power interruption impacted on our diagnostic and IT systems, which have now been fully restored.
"As a result, some outpatient clinics have been affected and one emergency patient was redirected to Hairmyres Hospital.
"This is in line with our business and emergency continuity plans which worked to good effect."
He added: "All patients affected by the interruption have been rescheduled and we would like to apologise for any inconvenience this will have caused."
A review of the Muslim Brotherhood's UK activity has cleared it of links to terrorism, its lawyers have said.
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A hospital was forced to reschedule operations after losing power for more than seven hours.