The scramble to buy GRE has become a two-horse race between Eureko, an alliance of seven European insurers and Royal & Sun Alliance.
The GRE board says it will be making a decision before the end of the month.
Royal & Sun is believed to value GRE at £3.4bn and has discussed offering 390p per share.
Eureko is understood to be prepared to match or better this with a split cash and shares offer that would take the share price to over 400p.
Previous favourite Axa has not withdrawn from the race but has made it clear that it is not prepared to offer more than 360p per share.