The news comes on the day that the ASA upheld a complaint by Morrisons that three Aldi adverts were misleading.
But Morrisons said that despite Wednesday's ruling, Aldi had launched another similar price comparison advert which was just as unfair.
Aldi said in response to the new complaint that it believes its new advert fully complies with ASA rules.
Morrisons latest complaint is about a price comparison TV advert tied to the Euro 2016 football competition. "It's clear they haven't changed their ways," a Morrisons spokesman said.
On Wednesday, the ASA ruled that two Aldi TV adverts and one press ad that claimed consumers could make significant savings were misleading.
One TV ad claimed a Â£70 Aldi shop would cost Â£98 at the "big four" supermarkets.
Morrisons and two members of the public complained the adverts did not make it clear Aldi's own-brand products were being compared with branded products.
However Aldi, said comparative advertising was a "well-established principle".
The German discount supermarket chain said consumers were likely to interpret the comparison as intended - the branded and fresh products from Tesco, Asda, Sainsbury's and Morrisons shown versus the Aldi products shown.
The ASA said: "We acknowledged that Aldi stated they had not intended the comparisons to represent a 'typical' weekly shop, but to be a comparison between the pictured products only.
"Nonetheless, we considered that was how consumers would interpret the adverts rather than as a representation of the savings which could be made by switching from a largely branded shop to shopping in Aldi, and therefore assessed them on that basis."
Aldi's UK and Ireland chief executive Matthew Barnes said it was "extremely disappointed" with the "ambiguous and inconsistent" decision.
"The use of comparative advertising is a well-established principle and is firmly in the interests of consumers and encourages competition between retailers," he said.
Supermarket Morrisons is to make a new complaint to the Advertising Standards Authority about discount chain Aldi.