The demand is the first sign of Senator Lambie's willingness to use her new position as a key swing vote in the Senate to extract benefits for her state,maximising her leverage as the fourth and possibly final crossbencher to back the tax plan.
Finance Minister Mathias Cormann responded to her demand on Wednesday afternoon by leaving room to negotiate with Senator Lambie and others.
"We are always happy to engage with Senators in relation to issues of concern to them and their constituents,"he said in a statement.
"We call on Labor and all other non-government senators to act in the national interest by supporting our income tax reforms in full."
Tasmania's social housing debt is equivalent to 0.1 per cent of the $158 billion value of the income tax cuts over 10 years.
The pressure to negotiate comes two weeks after the government signalled it would not do deals with the crossbench to pass the package.