The cars that can be traded in include any built to emissions standards that applied before 2010.
Vauxhall ran a similar scrappage scheme earlier this year, as well as in 2015 and 2016.
Analysis: Richard Westcott, BBC transport correspondent
No-one could accuse the car industry of rushing to solve air pollution.
Manufacturers have long been accused of dragging their heels over plans to tighten the legal emissions test.
And for years, they happily sold cars that they knew were far more polluting on real roads than in the official lab test.
But companies are now taking the initiative with old car scrappage schemes.
Ford is the latest to offer a plan, and their version insists that the older, polluting car is destroyed rather than resold.
But the offer is only open until the end of December.
And let’s be frank, it is also an attempt to boost sales which have been flagging across the industry for the past four months.
It’s hard to see it making a big dent in the dirty air problem.
Environmental lawyers’ campaign group ClientEarth welcomed Ford’s announcement.
“It seems the motor industry is finally waking up to the damage dirty diesels are doing to our lungs as well as their own reputation,” said ClientEarth lawyer Anna Heslop.
“What we need is a thought-through, coherent strategy from government to help people to move to cleaner and more sustainable technology.
“At the moment, there are pockets of small, short-term actions here and there, but nothing like the joined-up thinking we need to solve this problem.”
The UK government has come under pressure to announce a vehicle scrappage scheme for diesel cars, after it was found that air quality thresholds in cities were repeatedly being breached.
However the government’s clean air strategy announced in July did not include a scrappage scheme, calling previous ones “poor value” for money. Instead, it said new diesel and petrol cars would be banned from 2040.