New Chelsea boss Jose Mourinho wants an experienced top-class keeper to battle it out with Petr Cech for the No.1 shirt this season.
They had an initial bid turned down last week and an improved offer has now been similarly rejected by the Canaries.
Chelsea have released Hilario and Ross Turnbull, while the highly-rated Thibaut Courtois will spend a third season on loan at Spanish side Atletico Madrid.
That leaves youngster Sam Walker as the current back-up to Cech.
England international Ruddy started his career with Cambridge United before a high-profile move saw him sign for Everton in 2005.
However, he made just one appearance in five years for the Merseyside club and was loaned out to several teams including Stockport (twice), Crewe and Motherwell.
Having failed to make the breakthrough at Goodison Park, Ruddy opted for a move to Norwich in July 2010.
He has been one of the Premier League's most consistent keepers since, although he missed much of last season through injury.