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.

The benchmark Nikkei 225 closed down 3% at 16,883.19 points - its worst day since March.
The US oil price fell below the symbolic threshold of $50 a barrel for the first time since April 2009.
Meanwhile in London, Brent crude fell as low as $53 a barrel.
That prompted energy shares to fall.
Concerns over Greece's future also weighed on the euro, which slid to a nine-year low against the dollar.
Chinese shares recovered earlier losses after a private survey showed the country's services sector grew at its fastest pace in three months in December.
The Shanghai Composite ended up closing flat at 3,351.45 after falling as much as 1.3% earlier in the day.
In Hong Kong, the benchmark Hang Seng index finished down 1% to 23,485.41, following the global trend.
In Australia, investors also reacted to falling oil prices and eurozone worries, with the benchmark S&P/ASX 200 closing down 1.57% at 5,364.80 points.  Earlier in the day, the index was down as much as 1.9%  - one of its biggest falls in more than a month.
One company's shares saw a positive reaction to falling oil prices, however, with Qantas, Australia's national airline, seeing its share price at a four-year high during the Tuesday session.
Australia's latest trade figures showed the country's deficit widening in November from October to about 950m Australian dollars ($770m). It is the eighth trade deficit in a row.
But analysts said investors largely ignored the numbers, which were smaller than expected and due in part to falling prices for resources such as iron ore.
Tristan K'Nell, head of trading at Quay Equities, said there were a number of issues worldwide causing uncertainty in equity markets.
"Investors [are] watching closely... US interest rates, growth slowdowns in Europe and the Asia Pacific, political uncertainty in Greece, geopolitical issues between Russia and Ukraine and also the Middle East, and the volatility and crashing prices in crude oil and iron ore," he said.
In South Korea, the benchmark Kospi closed down 1.74% to 1,882.45 points, with shares in energy firms leading the declines on oil price worries.
Supermarket Morrisons is to make a new complaint to the Advertising Standards Authority about discount chain Aldi.
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Japan's stocks led falls across Asia on Tuesday, posting their biggest drop in nearly 10 months, as investors worried about falling oil prices and political turmoil in Greece.