The England keeper was the subject of a £5m offer from the Blues earlier this month, but the Canaries rebuffed that offer and have now rejected a second improved bid according to the report.
Chelsea are looking for a back-up to Petr Cech after Ross Turnbull and Hilario were released this summer. Thibaut Courtois, the highly-rated young Belgian international, has agreed to spend another season on loan with Atletico Madrid.
Jose Mourinho reportedly wants strong competition for Cech, and Ruddy - who was called up to the England squad in May 2012 after starring in the Premier League for newly-promoted Norwich - would provide just that.
After struggling to make an impression on the Everton first-team behind Tim Howard, Ruddy moved to East Anglia in 2010.
And he's been first-choice at Carrow Road for the majority of the time since, winning promotion with the club and consolidating their position in the top flight.
Although he missed much of last season through injury, the 26-year-old is higly regarded and young enough to act as understudy but talented enough to push Cech for a first-team place.
At the moment, youth team keeper Sam Walker is Chelsea's only reserve for the Czech Republic stopper.
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