Critics say the obligatory training – introduced after three pitbull terriers savaged and killed a six-year-old – has not achieved results, 20 Minuten reported.
It said Liberal National Councillor Ruedi Noser has brought a motion calling for the requirement to be lifted again.
Since 2008, there has been a legal requirement for around 300,000 dogs and their owners to attend two four-hour blocks of training and theory.
People who have previously owned dogs are exempt from the theory part.
But the authorities admit up to one-fifth of those who should attend do not – despite fines and the risk of prosecution.
According to the paper, Noser said the effort required to enforce the law was too great. He called instead for owners to take more responsibility for their dog’s behaviour.
The rightwing Swiss People’s Party (SVP) is also critical of the course requirement.
20 Minuten quoted SVP national councillor Sebastian Frehner as saying last summer that in introducing the courses parliament had “completely overreacted to a few tragic incidents”.
The animal protection organization Tier im Recht believes the course requirement should remain but that further training is required.
“Four hours is not enough for effective prevention (of attacks),” said spokeswoman Nora Flückiger.
The Federal Food Safety and Veterinary Office (FSVO) says it is possible that an analysis of cost and effect may lead to the course requirement being dropped.
But it says it is vital to have a central register of dog owners and completed training courses rather than leaving it up to the cantons.