The initiative, already deployed in other Italian cities including Turin and Genoa, comes as a bitterly contested deal to relocate hundreds of asylum seekers from Italy to other EU states has lagged.
In a plan put together by the city's authorities alongside the charity, Caritas, and the Italian Consortium of Solidarity (ICS), each family would have to host two asylum seekers for a maximum of six months and would be paid €400 a month for each person.
Some 700 migrants arrive in city, which is also a short distance from Croatia, each day.
“We are one of the gateways of the Balkan route," Roberto Cosolini, the mayor of Trieste, told La Stampa.
“Trieste, Udine and Gorizia: the whole region is having a significant impact from the flow [of migrants] who arrive by land.”
The city of around 200,000 residents is struggling to find accommodation for asylum seekers as boarding schools, hotels, serviced apartments and migrant centres reach capacity, prompting the appeal to local families to help.
“The families won't be left alone, they'll be assisted by Caritas and ICS," Laura Famulari, the councillor for social policies, said.
Needless to say, the anti-immigration Northern League was the first to take a swipe at the plan.
“It's crazy,” said Northern League politician Massimiliano Fedriga.
“This basically guarantees €400 per month for those who host an illegal immigrant in their home.”