The 41-year-old supermodel is rumoured to have split from The Kills rocker, and though he wants to take Staffordshire Bull Terrier cross Archie - who he rescued from Italy - with him to Los Angeles, the British beauty and her daughter Lila, 12, are devoted to the pooch and want to keep him with them.
A source said: ''So far things have been amicable, but the subject of Archie is really tricky and neither knows how to resolve the problem.''
Though Archie belongs to Jamie, Kate has always showered him with attention and gifts.
The source added to the Daily Star newspaper: ''Kate also loves Archie and even took him to a dog whisperer when he was having anxiety problems.
''She spends a fortune on his accessories and often walks him.''
The 46-year-old rocker has previously admitted to making long-distance phone calls to stay in touch with Archie when he is away on tour.
He is said: ''I get regular updates from Kate on how he's doing.
''He's pretty perfectly behaved. I speak to him on the phone sometimes.
''He came from the deepest, wildest countryside in Italy. He was almost feral. He needed me, so I adopted him.''
'Smalls Change (Meditations Upon Ageing)' arrives in April.
The two awards have made for a great 72nd birthday present for the country music icon.