The 15-year-old Welsh corgi-terrier mix - who accompanied the actress everywhere including on film location shoots - passed away a few weeks ago due to complications from old age.
Jennifer's representative said: 'He died a few weeks ago. He was an old dog and it was just his time.'
Jennifer - who recently bought a $4.95 million apartment in New York under the name Norman's Nest Trust - has previously been revealed to pay for Norman to receive a number of specialist treatments to deal with various ailments, including spending $250 a week on massage and acupuncture therapy.
A source revealed in the past: 'Norman has been Jennifer's constant companion during all her emotional upheavals, but he suffers from aching joints and stiffness. Jennifer doesn't want to put him on medication just yet, so she has opted for doggy spa treatments from a licensed vet technician.'
The 42-year-old actress - who has had a string of failed romances following her divorce from Brad Pitt in 2005 - has previously stated that she wished some of the men in her life had some of Norman's characteristics.
She said: 'It wouldn't be bad if, when a man comes home, he'd run to his woman with his tail wagging. This sort of excitement is something I've always missed in a man to be honest.'
Norman is survived by Jennifer's other pooch, a white German Shepherd named Dolly.