Andrew was arrested on Thursday on suspicion of misconduct in public office after allegations were made against him following the release of files related to pedophile billionaire Jeffrey Epstein.
Despite being stripped of his titles last year, the former Duke of York is still eighth in line to the throne and an Act of Parliament would be required to prevent him becoming king.
Keir Starmer's government will consider introducing such legislation once police have finished their investigation into the King's disgraced brother, the Press Association understands.
The Liberal Democrats would support the move to remove him from the monarchy's line of succession, and party leader Ed Davey told PA it would be "intolerable" for Andrew to succeed to the throne and that the scenario is "not as remote as some people think".
Any changes to the line of succession would require the agreement of other countries which share the UK monarch, including Australia, Canada and Jamaica.
It is also understood that an influential group of MPs will meet to consider launching a parliamentary probe into the role of UK trade envoys.
Any inquiry would start by focusing on potential governance issues within the wider system and lessons to be learned, avoiding specific commentary on Andrew - who spent 10 years in the role - until there is no risk of prejudicing criminal proceedings, PA understands.
This weekend the royal family will be attempting to conduct their duties as usual, with the Prince and Princess of Wales due to appear in public.
On Saturday the Princess of Wales wore an England Rugby scarf to watch the team take on Ireland in the Guinness Men's Six Nations at Twickenham, in her first public appearance since Andrew's arrest.
Andrew spent 11 hours in police custody on Thursday, on his 66th birthday, before being released under investigation, after allegations he shared sensitive information with Epstein during his time as the UK's trade envoy.
Detectives continued to search Andrew's former home, Royal Lodge, in Windsor, Berkshire, on Friday.
He has denied any wrongdoing over his links to the convicted sex offender, but has not directly responded to the latest allegations.
After his brother's arrest, the King said in a statement that "the law must take its course" and the police have "our full and wholehearted support and co-operation".
Charles attended the first show of London Fashion Week on Thursday, hours after Andrew's arrest, and did not respond when asked for his reaction to the news.
Detectives have asked Andrew's former close protection officers to "consider carefully whether anything they saw or heard" during their service could be relevant to the investigation into Epstein and his associates.
The Metropolitan Police said they were working with counterparts in the US to establish whether London airports had been used to "facilitate human trafficking and sexual exploitation".
On Friday, the force said no new reports of alleged sexual offences have been made since the release of millions of pages of documents related to Epstein by the US Department of Justice.
Scotland Yard previously said they were looking into allegations Andrew's protection officers turned a "blind eye" to his visits to Epstein's island, Little St James.
Police are yet to identify any wrongdoing by close protection officers.
Andrew's primary accuser, Virginia Giuffre, alleged she had sex with him during an orgy with "underage" girls on the Caribbean island.
The former prince has previously vehemently denied all allegations made against him.
The 14 Commonwealth countries where the King is head of state are Australia, Antigua and Barbuda, the Bahamas, Belize, Canada, Grenada, Jamaica, New Zealand, Papua New Guinea, St Kitts and Nevis, St Lucia, St Vincent and the Grenadines, Solomon Islands and Tuvalu.
