Police had earlier named British-born Salman Abedi, 22, as the perpetrator of the bombing at the Manchester Arena indoor venue at the end of a concert by US pop singer Ariana Grande on Monday, attended by thousands of children and teenagers.
"It is a possibility we can not ignore that there is a wider group of individuals linked to this attack", May said at her Downing Street office.
Officials are examining Abedi's trips to Libya as they piece together his allegiances and try to foil any new potential threats.
A 23-year-old man was also arrested in the south of the city in connection with the attack.
"Raising the threat level to critical will support our ongoing operation", he said in a statement.
Military reinforcements will be used to defend potential targets across the United Kingdom, allowing the police to "significantly increase the number of armed officers on patrol in key locations", May said. The traditional ceremony is a major tourist attraction in London.
Abedi also had ties to al-Qaeda, according to the official, and received terrorist training overseas.
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He was registered as living at Elsmore Road as recently as a year ago, where police raided a downstairs red-bricked semi-detached property on Tuesday.
Ramadan, who had not seen Abedi much in recent years, had noticed that Abedi had begun to dress "Islamically", in a long robe, and was growing a beard.
The BBC also reported that Abedi has at least three siblings and the family "has lived at several addresses in Manchester".
Officials are probing how often Abedi had travelled to Libya, which has seen an eruption of armed Islamist groups since dictator Moammar Gadhafi was overthrown and killed in 2011.
On his Facebook page, he took a swipe at Andy Burnham, stating that the Manchester mayor "says the attack is the work of an "extremist".
Minister Gerard Collomb said Wednesday on BFM television that British and French intelligence have information that Abedi had been to Syria.
According to ABC News in the US, British counterterrorism authorities had already identified Abedi as a potential threat, along with hundreds of others.
UK says Manchester attacker recently returned from Libya
Earlier Wednesday, French President Emmanuel Macron said he would seek an extension to the state of emergency until November. U.S. security sources, citing British intelligence officials, said Abedi was born in Manchester to parents of Libyan origin.
The Prime Minister added that the preliminary findings warranted an increase in the terror threat to the highest level.
May said she didn't want the public to be "alarmed", insisting the security response was "proportionate and sensible".
He said he also aided a girl who he said had lost her legs in the blast.
"Everyone was like scrambling over each other", she said. "Firstly, our thoughts and prayers go out to the victims of last night's tragedy, and due to the sheer number of people from the industry who have been called back to their desks to report on the incident, and the many attendees attempting to travel, we have made a decision to re-schedule the event for later this year".
Sixty-four people are still hospitalized, Jon Rouse of the Greater Manchester Health and Social Care Partnership said Wednesday.
A family friend, who asked not to be named, described him as "normal" and said they were known to the Libyan community in the city.
ISIS has previously told followers they will be rewarded "tenfold in paradise" if they die carrying out attacks during the Islamic holy month.
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Security consultant Michael Downing, a former deputy chief of the Los Angeles Police Department said entertainment venues have become standard "soft targets" for terrorists.