Negotiations between Athens and its eurozone and IMF creditors have dragged on for months.
"The deal's closure and the vote on measures (demanded by the European Union and the International Monetary Fund) will happen... on the condition that there be significant steps in the meantime on (reducing) debt," Tsipras told the Greek daily Ethnos in an interview published Sunday.
Negotiations between Athens and its eurozone and IMF creditors have dragged on for months over disagreements on debt relief and budget targets for the country.
Among the measures reportedly demanded by Greece's creditors are additional pension cuts, a reduced tax-exemption ceiling, and further deregulation of the energy and labour markets.
They are also pushing for a major asset sale at the state Public Power Corporation, Greece's largest electricity provider and a near monopoly, in the interests of increasing competition in the sector.
The impasse has held up the latest instalment of Greece's 86-billion-euro ($91 billion) bailout, agreed in 2015, which it needs for debt repayments in July.
"There will be new developments in the coming days," government spokesman Dimitris Papadopoulos told Greece's TV Saki on Sunday. "We are in the final stretch".
Representatives for Greece's creditors said progress had been made in certain areas but would not confirm that both sides were close to a deal.
The last such deadlock over Greece, which followed the election of Tsipras in early 2015, nearly saw Athens expelled from the euro.
With the European Union already reeling from Brexit, Tsipras argued that Greece's current rescue programme -- its third since 2010 -- would collapse if his government were to fall.
His leftist-led coalition has just 153 Ms in the 300-seat parliament, which would make it difficult for Tsipras to push through another package of austerity measures.