Regional officials estimate that up to two million Zimbabweans have sought economic refuge in neighbouring South Africa.
And critics of Robert Mugabe, the president, say the poor situation at home has caused a quarter of the country's 12 million people to flee.
The Sunday Mail said police in Plumtree on Zimbabwe's border with Botswana last week arrested and fined 24 people caught with fake travel documents, and the authorities were investigating the cases, suspected to be "part of a broader syndicate".
An official from the government passport office was quoted as saying: "Law enforcement agents in Zimbabwe and neighbouring countries have been advised to be on the lookout for those involved in the forgery."
Police and officials from the passport department were not immediately available for comment on Sunday.
The Sunday Mail said the passport office had received information that a criminal syndicate based in Bulawayo, Zimbabwe's second-largest city, was selling counterfeit papers for trips to Botswana, South Africa, Zambia and Namibia.
Fake papers are said to have
been used to cross borders
Hundreds of thousands of Zimbabweans make a living through cross-border trade - buying and selling commodities in short supply in their own country.
Critics blame Mugabe's government for an economic crisis that has left Zimbabwe battling frequent shortages of food, fuel and foreign currency, and with the world's highest inflation rate of nearly 1,200%.
Earlier this year, Harare's official Herald newspaper reported that about 100 Zimbabweans cross illegally into South Africa each day, risking drowning in a crocodile-infested river to search for jobs.