His announcement leaves both the leader and two co-deputy leader positions vacant. Under party rules,after a leader resigns there is a new leadership vote in 24 hours.
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Mr Bandt was deputy leader for three years under former leader Ms Milne,but lost the leadership race to Senator Di Natale in 2015. He has been co-deputy leader since mid-2019 and is the party's climate change spokesperson.
"I will be standing for Greens leader,"Mr Bandt tweeted in the wake of Senator Di Natale's resignation.
"I look forward to talking with my colleagues about how we share leadership across the House and Senate as we fight the climate emergency and inequality with aGreen New Deal (a sweeping "job-centred" plan to deal with climate change)."
Senator McKim immediately threw his support behind Mr Bandt,tweeting,"Adam is the right person to lead us".
While Senator Di Natale declined to name who he would back,he suggested there may only be one nominee for leader,telling reporters,"it could be a consensus decision".
Senator Di Natle was widely praised by current and former colleagues,with his predecessor Ms Milne saying she was"so very proud"and Senator Waters saying he had lead with"compassion and reason". Tasmanian Senator Peter Whish-Wilson added he brought"incredible energy to a difficult job".
Senator Faruqi thanked Senator Di Natale for his work,but appeared to be critical over his resignation clashing with with an ongoing push within the party to directly elect the parliamentary leader.
"Selecting our new leader is a great opportunity to democratically involve members in the process. The party should be doing just that for this ballot."
Senator Di Natale has backed the internal discussion about direct elections,but after a years-long debate,adecision on the issue is not expected until at least May.
Former Greens senator Andrew Bartlett,who has also been outspoken about the need for the direct election of a leader,said"people would be disappointed"by the timing of Senator Di Natale's announcement. But he added,"it's the reality of things".
"If Richard felt it was time to go,then just treading water in a role like that for another four months or so[until the direct election issue was decided],it's not desirable either."
Mr Bartlett also said there was no guarantee the party would ultimately decide to move to a direct election model.
Senator Di Natale plans to stay on as a Senator for Victoria until the middle of the year,when a replacement can be found.
The Victorian Greens will hold a ballot of its members in the coming weeks,after a brief internal campaigning period,to choose a replacement,with a wide field of candidates expected.
With Noel Towell