In an apparent about-turn after Kurz's previous administration with the far-right came crashing down, the 33-year-old said his People's Party (ÖVP) "will enter negotiations with the Greens".
But he warned of a "challenging process" ahead as the two parties with divergent platforms try to close gaps.
Green party leader Werner Kogler announced on Sunday that his party had approved the move to enter formal talks, bringing it closer to a potentially historic entry into government.
The ÖVP and the Greens made the biggest gains in a parliamentary election in late September.
The conservatives emerged as the strongest party, leading to Kurz being tasked to form another government.
If a coalition were to be formed between the two parties, it would be the first time for the Greens to enter government in Austria at the national level.
Kurz, whose party got 37.5 percent of the vote, said he could not tell how long negotiations would take, saying the two months it took to reach agreement with the far-right Freedom Party (FPÖ) in 2017 had been "extremely fast".
"The process won't be easy because the positions of the Greens and the ÖVP are very, very different," Kurz told reproters, citing his party's promises to be tough on migration and boost the country's economy.
Kogler, whose party takes a more liberal line on immigration, has stressed too that any agreement would "necessitate many compromises".
The ÖVP and the Greens previously entered coalition negotiations in 2003 but on that occasion they failed and the ÖVP entered government with the far-right instead.
Kurz's previous government with the FPÖ was brought down in May when the so-called Ibiza-gate corruption scandal engulfed his ally after just 18 months in government together.
The FPÖ suffered heavy losses during the September election, while the Greens made big gains due to the environment becoming the top voter concern, securing 13.9 percent of the vote in their best results ever and after failing to get into parliament in 2017.