The opposition centre-left Labour Party and minor Greens party used their Senate majority in March to block the bills to remove the tax, which was introduced by a Labour government in July 2012.
But with new senators taking their seats on July 7 for the first time since Abbott's conservative coalition government won elections in September, the bills are expected to be passed by a narrow margin.
While introducing the bills to the House of Representatives where the government holds a clear majority, Abbott said today that voters had rejected the tax by electing his government.