The supermarket is targeting people who regularly shop at the store, but normally do their main festive shop elsewhere, using information taken from the company's Clubcards.
Customers have been sent a£5 gift card and a voucher offering them£18 off when they spend£70.
Competitors have described the offer as 'madness', saying it is likely the retailer will lose money.
An Asda spokesman said: 'We are surprised they are doing such an aggressive and targeted price-cutting, given the concerns of the Competition Commission about local pricing.'
The Competition Commission is investigating the grocery market at present, questioning retailers about price promotion offers.
However, a Tesco spokesman said: 'This is a national, targeted promotion. We know that a small number of customers shop with us year round but then, for one reason or another, try somewhere else at Christmas.
'We hope that this promotion will encourage them to try some of our fantastic Christmas products at great prices.'
A sales leap of 17.9 per cent at Tesco's overseas division in the 13 weeks to November 25 lifted group sales, excluding petrol, by 10.3 per cent over the period.
In the UK, like-for-likes rose 5.6 per cent - slightly ahead of some analysts' expectations - on total sales up 8.5 per cent.
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