Labor is on track to win crossbench approval for curbing tax concessions for Australians with more than $3 million in superannuation, after Tasmanian Senators Jacqui Lambie and Tammy Tyrrell joined a growing bloc signalling support.

But the move to impose higher taxes on funds well above the national average might not go far enough for the Greens, who could push the Albanese government to lower the cut-off for concessions to $2 million.