Heaney has taken on the former Arbennig restaurant site – which closed last month – and the next door Emporium, in the Pontcanna area of the Welsh capital.
He is aiming to raise £40,000 to help refurbish the sites, with the aim of opening later this year.
Rewards range from tickets to guest chef nights from the likes of Richard Bainbridge and Tom Brown, to the chance for one backer to have their initials tattooed on Heaney’s arm.
The chef will choose the font, but he promises each letter will be “larger than a 10p piece and will be somewhere on his arm” with “maximum three initials”.
The restaurant will serve fresh, simple food in an “informal” 50-cover main dining room and cocktail bar, with “fuss-free” service.
A smaller Deli area next door will offer small plates, private dining and drop-in lunches. Heaney hopes to use the space to experiment without impacting the atmosphere of the main restaurant.
The Kickstarter campaign began on 20 June, and at the time of writing has raised over £9,000 - though the tattoo reward is still unclaimed.
Writing on Twitter Heaney thanked his supporters for “an amazing first 24 hours”.
The chef also runs Restaurant Tommy Heaney at The Great House in Brigend, South Wales, where head chef Garth Denyer will run this kitchen in his absence.
Heaney also made it to the final of the BBC’s Great British Menu in 2017 and is returning to represent Northern Ireland in the next series, which airs this summer.
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