Rouhani, a moderate elected last year, has already drawn fire over a landmark nuclear deal with world powers and diplomatic overtures that have been broadly welcomed in the West but slammed by hardliners at home.
Both the diplomatic push and the social programme are aimed at alleviating the effects of crippling Western sanctions imposed over Iran's nuclear programme, which the West suspects is being used to develop weapons, charges denied by Tehran.
The programme aims to hand out packages of frozen chicken, rice, vegetable oil, cheese and eggs to poor families ahead of Iranian New Year celebrations in mid-March.
But critics say the programme has forced people to stand in long queues in cold weather, harming the image of the Islamic republic, and local media reported that an elderly man had died while waiting in the western Kermanshah province.
Parliament speaker Ali Larijani has urged the government to "immediately amend the distribution method," the Khorosan daily reported.
The hardline Kayhan newspaper said Rouhani should apologise over the programme and called for officials involved in it to resign.
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"This plan could have been a gesture that the government cares about the poor … but it has turned into a damaging measure accompanied by insult and humiliation," it said in an editorial.
"Making huge numbers of poor people wait in long lines in freezing weather … has been used by enemy media to portray a dark image of Iran."
Even Reformist newspapers criticised the government for what they said was a hasty decision to implement the scheme without proper preparation.
In response, Rouhani has tasked Welfare Minister Ali Rabii with re-organising the handout effort and improving distribution, media reports said.
Rabii has meanwhile defended the plan, saying at least nine million baskets of goods would be distributed.
"The policy of this government is to provide welfare for the whole society," he said in response to those he accused of "unfairly" attacking the programme, the official IRNA news agency reported.
"Unfortunately, these people have no perception of national interests."