The amateur rider fractured two vertebrae in his neck during a fall last Wednesday and had to have surgery on his broken neck.
"JT McNamara remains stable in an induced coma in Frenchay Hospital, Bristol. Further scans are planned and another statement will be issued as soon as there is more news," said a statement issued jointly by Dr Adrian McGoldrick, senior medical officer for the Irish Turf Club, and Lisa Hancock, CEO of the Injured Jockey's Fund.
"Whilst visits to JT at the moment are not appropriate, JT's family are extremely grateful for all the kind messages and support."
The 37-year-old was riding Galaxy Rock for trainer Jonjo O'Neill in the Fulke Walwyn Kim Muir Challenge Cup on Wednesday 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.
AP McCoy, who has been champion jockey 17 times, told the BBC at the time: "It's a dangerous sport and it's a sight you see that unfortunately crops up.
"We're very lucky we have excellent medical care. We just hope he's going to be OK."
McNamara is a leading Irish amateur rider who has won four races at the Cheltenham Festival in his career.