Garff MHK Martyn Perkins said the public are "sick of dog mess dirtying the streets and popular walks".
A two-week consultation will be launched by the Department of Environment, Food and Agriculture to gauge public opinion.
Currently, owners can be given a fixed penalty fine of Â£50 or taken to court if they don't clean up after their dog.
The proposed system will see swabs taken from dogs' mouths and compared to samples taken from faeces.
Mr Perkins said he had "already had lots of feedback from responsible owners who support the move", adding: "It is certainly a case of the minority ruining it for the majority."
He said the issue needs to be dealt with partially because the island's "outstanding natural landscapes" have recently been recognised by UNESCO.
"We are very proud of this status and we want to encourage people to take pride in their environment," he said.
If it is supported, DNA testing could be included in a bill, currently going through Tynwald, which aims to amend the Dogs Act 1990.
The bill would mean every dog on the island must be implanted with a microchip, which would see an end to the licensing and duty regime for dogs.
Microchipping became a legal requirement in the UK in 2016.

MoneyGram has about 350,000 outlets in nearly 200 countries. Ant Financial has more than 630 million users.
The takeover by the Chinese group will need regulatory approval from the US Committee on Foreign Investment.
The inter-agency committee reviews foreign acquisitions of domestic American assets on grounds of national security.
Eric Jing, chief executive at Ant Financial, said in a statement that the marriage of the two companies will "provide greater access, security and simplicity for people around the world to remit funds, especially in major economies such as the United States, China, India, Mexico and the Philippines".
Ant Financial has a big market share in the online payments industry in China. The acquisition could help the company extend the lead as well as expand overseas, as competition is growing in China with rival Tencent's WeChat payment system.
US-listed Moneygram's shares rose by nearly 9% on the news. The takeover has been approved by MoneyGram's board of directors.
Ant Financial's shopping spree in the US comes against a backdrop of rising tensions between China and the world's biggest economy.
Before he took office, then president-elect Donald Trump was questioning whether the US should continue its "One China" policy, sparking fury from Chinese state media. And during his presidential campaign, Mr Trump threatened to impose punitive tariffs on Chinese imports.
But Jack Ma, the founder and chairman of Alibaba, held a meeting with Mr Trump in December last year.
While President Trump has been critical of China, he said he had a "great meeting" with Mr Ma, who chose to float Alibaba on the New York Stock Exchange. The share sale in September 2014 was a record-breaker, as Alibaba raised $25bn in its initial public offering.
If the MoneyGram deal goes through, it will be Alibaba's second acquisition in the US. Last year the e-commerce giant purchased EyeVerify in a $70m deal.
EyeVerify is a start-up based in Missouri, which uses biometric authentication technology for securing user's online data and transactions.
DNA testing of dog poo has been backed by a Manx politician who wants it used to tackle fouling on the Isle of Man.
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Ant Financial, the digital payments arm of e-commerce giant Alibaba, is buying US-based MoneyGram for $880m (£700m).