Age Scotland says official figures suggest that each year about 8,640 people in Scotland wait longer than six weeks for a council care assessment.
The average "worst case" is between two weeks and five months, with the longest delay of 18 months recorded in 2014/15.
Cosla, the umbrella body for local councils, declined to comment.
Age Scotland's research, compiled using freedom of information requests, found significant variations across councils in the time taken to carry out assessments.
Under national guidelines, people should wait no longer than six weeks for care services to be provided after an assessment has taken place.
Age Scotland found that about 10% of people were not provided with services within the six-week limit over the past three years.
The charity said its research suggested that about 3,000 elderly people each year wait longer than six weeks for services to be put in place.
Most councils do not record the reasons why delays occur, but many cited instances where delays were caused by the person being admitted to hospital or waiting for a place in their chosen care home.
Staff shortages, financial constraints and delays in adapting homes were also cited.
Age Scotland received freedom of information responses from 25 out of 32 councils.
Chief executive Keith Robson said: "These are deeply concerning figures showing thousands of older people facing delays in the care provision they need being put in place.
"It also means payments for free personal care they are entitled to are not being received.
"This confirms the experiences of a number of older people and their families who have been in touch with Age Scotland's helpline to tell us their experiences of delays in the system.
"As we look to local authority elections next month, Age Scotland has contacted council candidates across Scotland to ask them to ensure providing high-quality health and social care services is made an urgent priority by new administrations."
Thousands of elderly people are missing out on free personal care because of delays to assessments and care arrangements, a charity has claimed.