An 86-year-old woman was taken to hospital after being found by family members injured within her address at Kirkhill Drive, Luncarty, at about 9.30pm on Friday, May 28.
She remains in hospital receiving treatment.
Officers are seeking to trace anyone who may have attended her address in the days leading up to when she was found.
Police are especially keen to trace a man who is believed to have attended her address in the week days prior to Friday.
He is described as a white man in his fifties, with short spiky grey hair and grey stubble. He was wearing glasses, a black fleece and dark trousers and was carrying a black folder.
Detective Inspector Marc Lorente from the Specialist Crime Division said: “A significant amount of enquiry has been carried out since this incident was reported and how the woman came to be injured is still unclear.
“Our enquiries are focusing on people who may have been in the Kirkhill Drive area of Luncarty during the week commencing Monday, 24 May, including the described man.
"The address is at the junction with the main road through Luncarty and we’d ask anyone who noticed any unknown vehicles in the area, at the address or in the vicinity, to get in touch.”
Anyone with information can contact 101 and quote Operation Jackdrop, or incident 2847 of May 29.
Information can be given to Crimestoppers on 0800 555 111.