The 27-year-old, who has long been linked with a move to the Premier League, is set to sign a three-year contract at the Liberty Stadium, Sky Sports understands.
The Swans' accepted offer is an improvement on their initial bid of ?4million, (5m euros) which was not enough to lure the Spain international away from the Mestalla.
Hernandez has a contract at Valencia until 2014 and coach Maurizio Pellegrino was keen to keep him at the La Liga club.
But Swansea appear to have finally got their man, who has previously been linked with Manchester City and Liverpool.
The Welsh side are yet to confirm the agreement but reports in Spain suggest a move is imminent.
The switch will see Hernandez reunited with manager Michael Laudrup, who coached him during their time together at Getafe.
