Local media have reported that the three died in recent days while standing in line in freezing temperatures. Authorities were quoted as saying that they had pre-existing heart problems.
Most provinces in Iran have experienced unusually low temperatures in recent days.
Rouhani told state TV late Wednesday that he "as the president expresses regret if people have faced trouble in receiving the commodity basket."
The ration for the poor includes eggs, cooking oil, chicken, rice and cheese. The program was instituted under Rouhani's predecessor, Mahmoud Ahmadinejad, originally just for government workers.
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Rouhani's administration decided to extend it from 3 million people to 17 million, out of a total population of 76 million.
The plan is largely intended to counteract inflation, fed by the sanctions imposed on Iran over its nuclear program. Abolhasan Firoozabadi, the deputy labor minister who handles the plan, told the official IRNA news agency that more than 11 million baskets of goods have been distributed so far.
Ramin Pakzad, a part-time worker, said he tried and failed to find a line short enough that he wouldn't risk losing his job by standing in it.
Rouhani's opponents have leaped on the failures of the plan, with the hard-line Javan daily declaring it a "national humiliation."
But even many of his supporters also acknowledge its shortcomings. "I think this is an uncalculated idea that government applied hastily and without any precautions," said Sadegh Zibakalam, a Tehran University politics professor who supported Rouhani in June presidential elections.