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."

Wayne Maycock, Paul Bromwich and Admi Headley were last seen at HMP Leyhill at about 16:45 GMT on Sunday.
Avon and Somerset Police has appealed for anyone who sees them, or knows of their whereabouts, to make contact.
HMP Leyhill in South Gloucestershire is the only minimum-security prison in the South West.
Headley was sentenced in 2006 for rape and robbery, Maycock was jailed in the same year for GBH, while Bromwich was sentenced in 2001 for rape.
Earlier, the police force suggested all three were convicted of rape but this information was later amended.
A Prison Service spokesperson said: "Public protection is our top priority. We take absconds from custody extremely seriously. We are working closely with the police and are urgently investigating this incident."
According to Avon and Somerset Police:
Leyhill, near Wotton-under-Edge, is a category D prison housing more than 500 inmates, including some on life sentences.

The emails confirmed the Trump camp's interest in receiving help from the Russian government to discredit opponent Hillary Clinton during the presidential campaign last year.
The Dow Jones was unchanged at 21,409.07 and the S&P 500 edged down 0.08% to 2,425.53.
The Nasdaq rose 0.3% to 6,193.3.
Wall Street has generally climbed this year, in spite of some shaky moments stemming from unease over the fate of the Trump administration and prospects for its business-friendly agenda.
Markets also sank in May amid a furore over US President Donald Trump's firing of former FBI Director James Comey, who had been investigating ties between the campaign and Russia.
That investigation is ongoing.
On Tuesday, a rise in oil prices helped lift energy stocks, such as Exxon Mobil, which climbed 0.6%.
But bank stocks lost ground ahead of the release of quarterly earnings this week and a day after the US Consumer Financial Protection Bureau unveiled a rule making it easier for customers to file collective lawsuits.
introduced a The rule forbids companies from requiring that people use an individual, out-of-court process to resolve disputes as part of the contract for their credit card and bank accounts.
Bank of America and Citigroup shares fell 1.2% on Tuesday, while Wells Fargo dropped 1%.
The rule may be short-lived as it comes amid a push by the White House and Republicans in Congress to make the Consumer Financial Protection Bureau less powerful.
Senator Tom Cotton, a Republican who represents Arkansas, said on Tuesday he would file legislation to undo the rule.
Thousands of elderly people are missing out on free personal care because of delays to assessments and care arrangements, a charity has claimed.
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Two convicted rapists and a man convicted of assault - all considered to be a "risk to the public" - have absconded from an open prison.
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The US stock market closed on Tuesday about where it began, recovering after a flash of panic prompted by the release of emails from Donald Trump Jr.