The amateur rider fractured two vertebrae in his neck during a fall last Wednesday and had to have surgery on his broken neck.
A statement issued on behalf of the McNamara family read: "JT McNamara remains in the Frenchay Hospital, Bristol. Whilst he suffered a serious neck injury resulting in paralysis, he has made progress in the last week and is in a very positive frame of mind.
"He is greatly appreciative of the many messages, cards and letters received and also wishes to thank the Frenchay Hospital who are looking after him so well."
McNamara has escaped brain damage, and a statement issued last week confirmed that he is fully alert but requires full-time ventilation. He is expected to be transferred to the National Spinal Unit in Mater Hospital, Dublin, in the coming weeks. His wife and three children are all based in Ireland.
The 37-year-old was riding Galaxy Rock for trainer Jonjo O'Neill in the Fulke Walwyn Kim Muir Challenge Cup on Wednesday of the Festival when the pair fell at the first fence.
An air ambulance landed at the racecourse to treat the injured rider, who was conscious after the fall but had difficulty breathing, and he was taken to the hospital.
As an amateur rider McNamara's career included four winners at the Cheltenham Festival.
- Sports & Recreation