Allt Na Reigh, in Glencoe, was targeted at the weekend by vandals, who reportedly sprayed it with orange paint.
Last month, Northern Constabulary confirmed that “abusive slogans” were painted on the walls of the property after child and sex abuse allegations against the former Top Of The Pops and Jim’ll Fix It host emerged.
The words “Jimmy The Beast” were painted across the front of the building while “Beast” was sprayed over a side wall.
Officers have searched the cottage to look for “any evidence of any others being involved in any offending with him”.
But the Metropolitan Police refused to comment on reports that more than 20 allegations of abuse are being investigated there.
Plans to turn the remote property into a respite care centre for the disabled were halted when the Jimmy Savile Charitable Trust closed down in the wake of the growing scandal.
Highland councillor Andrew Baxter, who represents the Fort William and Ardnamurchan ward, said: “I think it (demolition) is a final option. It is certainly one that I have heard suggested by local residents.
“The general feeling amongst the community is that we would want to distance ourselves from Jimmy Savile and the fact that he lived here and perhaps perpetrated some of his vile crimes here, and move on.
“The cottage is in such a prominent position, it’s been on the front page of many newspapers and it is becoming a focus for a lot of mindless vandals.
“We don’t want to be reminded of what might have happened there and we don’t want visitors to Glencoe being reminded either.
“But demolishing should be a final resort. One of the other options being suggested is that it could be turned into a mountain centre, as a base to get people - possibly those who are disabled or from disadvantaged backgrounds - out to the mountains.
“However, my feeling, and the feeling of others, is that it may not be appropriate to use it as overnight accommodation, considering what may have happened there.
“As I understand it, the cottage is still owned by the Jimmy Savile Charitable Trust, but I have no idea what they are planning to do with it.”