One line of inquiry is expected to examine whether the five deliberately sparked the alert to land in the UK.
The aircraft, travelling between Egypt and the United States, was diverted to Prestwick in Ayrshire on Saturday after a threatening note was found on board.
The asylum seekers remained in the UK after the plane was given permission to take off.
Police have refused to rule out a link between the terror alert and the asylum seekers, but said their investigations were continuing.
A spokeswoman for Police Scotland said five of the plane’s passengers are now being dealt with by the UK Border Agency.
Tawfeek Asi, chief executive officer of the plane’s operator Egyptair, identified the group as Syrian.
The incident came as Prime Minister David Cameron met Russian president Vladimir Putin in London yesterday to discuss the deepening Syrian crisis.
The US is preparing to arm rebels, in response to the Syrian government’s alleged use of chemical weapons against its own people, with speculation that the UK may follow America’s lead.
Mr Putin, meanwhile, has said he will supply president Bashar al-Assad with anti-aircraft missiles in order to deter “hotheads” from intervening in the conflict.
Police Scotland would not comment on whether the five asylum seekers are now suspects in connection with the note found on the flight, which said: “I’ll set this plane on fire”. However, there were calls for any possible link to be investigated.
Last night, the Home Office would not comment on the passengers seeking asylum in the UK. No arrests have been made.
A Police Scotland spokeswoman said: “We are continuing to investigate the circumstances.”
Flight 985 was heading to New York from Cairo when BBC producer Nada Tawefik found the threatening note, scrawled on a napkin, in a toilet when attempting to change her baby.
It had the seat number 46D written on it.
The note was passed to crew members and the plane was escorted to Glasgow Prestwick, which is designated to deal with emergency incidents, by Typhoon fighter jets from RAF Leuchars.
The Boeing 777 was grounded at 2:30pm on Saturday, 326 passengers were evacuated and the plane was searched for six hours.
Nothing was found and the aircraft was cleared to resume its journey early yesterday morning.
The five passengers who have claimed asylum remained in Glasgow and were yesterday believed to have been taken to have their bids assessed.
The Home Office said that each claim is considered on its merits, and that the UK has a proud record of granting asylum to those who need it, while also ensuring that those who do not are returned to their own country.
Lewis Macdonald MSP, Scottish Labour justice spokesman, said it should not matter that the plane was not initially
destined for Scotland.
“At the end of the day, it does not matter how someone pitches up on British shores – if they have a case for asylum the law allows them to make it,” he said.
However, he added that any possible link between the alert and the asylum claims should be investigated rigorously.
“It would be an obvious and logical thing for the police and the borders authority to work together on, and get to the bottom of,” Mr Macdonald said.
“If they find any connection, they should be prosecuted in a Scottish court and then removed from the country if they are found guilty. This is a very serious offence”
It is not clear why the five chose to make their asylum claim in the UK, rather than the US.
The two countries take a similar approach to asylum, restricting it to people facing persecution in their home country, and in some cases detaining applicants during the process.
The US, which takes in more asylum seekers than any other country, has a blanket ban on terrorists being approved,
while the Home Office’s criteria make no reference to terrorism, but do take criminality into account.
Saturday’s alert was the most high-profile security threat Scotland has faced since the 2007 attack on Glasgow Airport.
It came after weeks of heightened tensions, following the murder of soldier Lee Rigby in Woolwich last month.
Within days of that attack, RAF Typhoons were scrambled to escort a Boeing 777, which was travelling from Pakistan to the UK, after the pilot sent out an emergency code.
The Manchester-bound flight, carrying 297 passengers, was
diverted to Stansted, and two men were arrested and taken into custody.
Despite this, the terror level has not been raised, and Prestwick Airport staff said they were pleased with the way Saturday’s threat was handled.
Iain Cochrane, chief executive, said: “We carefully plan and train for this kind of eventuality, and I am relieved it ended safely and without incident.
“It was a complex and constantly evolving matter, and I want to thank my staff for their unswerving professionalism and commitment during the 13 hours the aircraft was here, and praise the excellent manner in which all agencies involved worked together to resolve the matter.”