Since January 2008, a law passed by the right-wing majority in the cantonal parliament has meant beggars can be fined if caught asking for cash.
But Catholic charity confederation Caritas said the measure had not proved effective in deterring mendicants.
"The phenomenon has not diminished, and prohibiting it leads to considerable costs for taxpayers, both in terms of police personnel and administrative and judicial fees," said Caritas Geneva in a statement.
According to the coalition, fining beggars is inhumane, discriminatory and stigmatizing. It is also very expensive, it says. In 18 months, the cost of filing some 13,634 reports has amounted to three million francs ($3.25 million).
According to the coalition, the fact that the fines had been issued to a total of 1,516 people showed that there was a high rate of recidivism and the policy was having no effect.
"The criminalization of people in a precarious situation, principally Roma, leaves room for all sorts of abuses against them (confiscation of the money collected, or of their belongings, destruction of their makeshift shelters, etc.), further aggravating an already fragile situation," said Caritas.
The group hopes to collect 10,000 signatures by March 31st before submitting its proposal to the cantonal parliament.