Shares in G4S – which runs UK prisons and botched security arrangements at the London Olympics – closed nearly 20% higher after a Canadian rival admitted it was considering a takeover.
Garda World, a privately-owned company, has until 8 May to table a formal offer.
There was no immediate comment from G4S, which last month reported a 63% fall in full-year profits.
G4S was valued at more than £3bn after the rise in its share price.
The business – which has operations in 90 countries and employs more than 540,000 – also provides staff for events and collects and transports cash in armoured vans.
Its largest geographical region is Asia, where 32% of its employees work.
But it is often embroiled in controversy. Last week the company was stripped of the contract to run HMP Birmingham, which was awarded in 2011 for 15 years.
That came after it was forced to transfer the management of Medway Secure Training Centre to the government in 2016, after secret BBC filming showed staff allegedly mistreating children held there.
In the UK, its reputation took a knock in 2012 when it had to admit, 16 days before the London Olympics began, that it had not recruited enough security staff for the event.
Then in 2014, G4S agreed to repay £108.9m plus tax to the UK government after overcharging on contracts to tag offenders.
Garda World – which describes itself as the world’s largest privately owned security company – did not disclose many details about its possible offer.
But it said in a statement, sparked by a report in London’s Evening Standard newspaper, that it was “in the preliminary stages of considering an approach to the board of G4S regarding a possible cash offer for the company, or a part thereof”.
G4S has admitted in the past to receiving unsolicited interest for its cash handling division.
In December last year, it had said it was looking to demerge or list the cash unit.