"Ukrainians, if you need me to be president, I'll be president," Savchenko told journalists in Kyiv.
Ukraine's first female military pilot, Savchenko was appointed to the parliamentary assembly of the Council of Europ, based in Strasbourg, during nearly two years in captivity, and some have wondered if she will be the country's next leader.
In a news conference, she refused to rule this out.
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The 35-year-old said that what she would like best is to return to her job as a military pilot but is willing to launch a political career if this could help Ukraine deal with the conflict with Russia.
Meanwhile, MEPs from across the political divide have welcomed her release and return to Ukraine.
Savchenko had been detained and moved to Russia by pro-Russian militants in 2014.
She was swapped for two Russian prisoners, Yevgeny Yerofeyev and Alexander Alexandrov, who were held by Ukraine, and flown from the Russian city of Rostov-on-Don to Kiev on the Ukrainian President's personal plane.
On Monday, Polish MEP Jacek Saryusz-Wolski, a Vice-Chair of Parliament's EPP group, described the pilot as "the Ukrainian Jeanne d'Arc."
"She has been released after two years of illegal imprisonment. This success is a result of the fight of the Ukrainian and international communities, including the European Parliament," said the deputy, who is responsible for foreign affairs in the EPP group.
Further reaction came from Latvian member Sandra Kalniete, who said, "I strongly believe that the perpetrators of the crimes against the Ukrainian people should be, and will one day, be held accountable under international criminal law."
Romanian EPP group MEP Cristian Dan Preda, meanwhile, called for the release of all Ukrainian political prisoners in Russia, saying, "The EPP group and Parliament have repeatedly demanded for the last two years the release of Savchenko, who was illegally detained by the Russian authorities.
"While we are happy to welcome her release and return to Ukraine, I nevertheless call once more on the Russian authorities to release all other Ukrainian political prisoners and to respect the territorial integrity and sovereignty of Ukraine".
ECR group member Mark Demesmaeker has been pushing for the release of Savchenko ever since her capture and he was "happy" that the deal has finally been reached.
The Belgian deputy said, "Savchenko should never have been in prison. She was condemned without any clear evidence. Her imprisonment was a political decision. But on the other hand, we should also be critical towards the conviction of the Russian soldiers."
In March, Savchenko was convicted of acting as a spotter for mortar fire that killed two Russian journalists and sentenced to 22 years in a Russian prison. She denied the charges and was pardoned on humanitarian grounds by Russian President Vladimir Putin.
The Russian soldiers were sentenced to 14 years in jail last month after being found guilty of committing a terrorist act and of provoking an armed conflict in Ukraine.