Immigration New Zealand said it was investigating the two visitors and would consider cancelling the visas.
"Under the conditions of these individuals' two-year multiple entry visitor visas, their visa expires one month after each arrival," said agency spokesman Marc Piercey.
"As part of our investigation a decision will be made on whether there are grounds to cancel their visas when they first leave New Zealand to prevent them from returning here."
The two men had been given a police warning for allegedly breaching Auckland Council bylaws.
According to identity documents sighted by the Herald, one of the men is a Chinese national named Chen Ming Xin.
The 38-year-old arrived in Auckland on December 31 and is therefore required to leave the country by the end of this month.
Visitor visa holders are not allowed to undertake any employment or work while in New Zealand.
"Visitor visa holders must be coming to New Zealand for lawful purpose such as holidaying, sightseeing, family and social visits and playing amateur sport," Mr Piercey said.
"Work means any activity undertaken for gain or reward and includes any payment or benefit that can be valued in terms of money, such as board and lodging, goods and services."
Begging or soliciting donations could also be considered to be an activity undertaken for gain or reward, Mr Piercey said.
Police have thanked the Herald for the coverage of the matter, but said it would not be making any further comments at this point.
"Thank you for your coverage of the matter to date," Auckland Central acting area commander Gary Davey said in an email.
"It is important to make members of the public aware of this type of behaviour."
On Monday, a woman was left "annoyed and afraid" after a monk got angry when she refused to pass him cash.
Another woman was last week pressured into giving a monk $10 after he followed her into a Queen St shop and asked her to get "cash out" from her Eftpos card.
Police have asked the public to stop giving money to beggars on Queen St, and instead refer them to agencies such as Lifewise and City Mission.
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