RSA said Zurich had made a revised approach worth 550p a share, which comes just hours before a 5pm deadline for Zurich to make a firm offer or walk away under City takeover rules.
RSA has now asked for the UK Takeover Panel to extend the deadline until September 22 as they hammer out a formal deal.
Details of the proposed bid comes after Zurich recently insisted it would not overpay for RSA.
But a price of 550p a share will no doubt come as a relief to investors after speculation Zurich was working on a lower-than-expected offer of under 530p a share.
RSA said its board had “indicated to Zurich that it would be willing to recommend an offer at the level of the possible offer to RSA shareholders”.
Shareholders will also receive the dividend payout worth 3.5p a share if the takeover goes ahead, lifting the overall price to £5.63 billion.
Insurance expert Eamonn Flanagan at Shore Capital said a takeover for RSA was now “inevitable”.
But he added a rival offer was still possible “from another Continental European insurer”.