The opposition and supporters have taken over several government buildings, including the presidential administration and parliament.
It was reported late on Friday that Mr Bzhania had met with protesters and agreed to step down if they appointed vice-president Badr Gunba as the region’s leader.
The opposition is understood to have rejected this offer and is demanding the resignation of the entire presidential team, saying all of them have “compromised themselves”.
The president previously championed the deal, which sought to lift the region’s ban on the foreign ownership of residential property,
With no international investment entering Abkhazia – because the territory is considered illegally occupied by most UN member states – the new bill was expected to largely benefit Russian investors.