Allied Universal has offered 245p or 10p per share more for the FTSE 250 security and guarding group.
The US group has also told the UK Takeover Panel that this offer is final and will not be revised while it has reduced the acceptance requirement for its bid to 75% from 90%.
The Allied Universal offer values G4S at approximately £3.8bn.
"Allied Universal’s final offer of 245p per share reflected the transformation of G4S under [CEO] Ashley Almanza and his leadership team, supported by the impressive efforts of all G4S employees," said John Connolly, G4S Board chairman.
“The business has been fundamentally refocused and re-positioned as a global leader to address long-term growth in security markets, with a unique market footprint and industry-leading expertise.
“We are therefore unanimously recommending it to our shareholders,” he said.
Allied Universal and GardaWorld agreed to a sealed bid auction process to determine the fate of G4S but with both bids now final it is up to shareholders to decide.
The Takeover Panel has confirmed that, as agreed in the auction rules, the latest time by which either offeror must have satisfied the acceptance condition of its offer is 13.00 on 16 March 2021.