Government spokesman Zoltan Kovacs said that Interior Minister Sandor Pinter assured a Cabinet meeting that the deadline, which had earlier been set for Nov. 30, could be met.
According to Kovacs, Pinter said "that the security fence can be built along its full length in a month."
Close to 100,000 migrants and refugees have entered Hungary so far this year, compared to less than 43,000 asylum seekers in all of 2014. Over the past few weeks, especially since Hungary built a sample section of the fence on the border, the number of arrivals has gone up, often exceeding 1,500 people a day.
Kovacs said the government had earmarked 22.2 billion forints ($79.3 million) to fund the 175-kilometer (109-mile) long fence, in addition to 6.5 billion forints set aside earlier.
Some of the funds will be used also to set up tent camps in rural areas for the migrants and refugees, gradually replacing housing in cities and towns.
About 80 percent of the asylum seekers are now from war-torn countries like Iraq, Syria and Afghanistan. Most try to reach destinations further west in the European Union, like Germany or the Netherlands, before their asylum claims are settled.
A government campaign rejecting migration and linking it to terrorism has been strongly criticized by the U.N refugee agency and others.