Iranian Aideen Strandsson, 37, faces prison if she is returned to her home country because she was once a Muslim.
According to Christian Broadcasting Network, Swedish officials told Strandsson "It's not our problem if you decided to become a Christian".
Strandsson's lawyer Gabriel Donner said: "Torture and rape is common in Iranian prisons."
Although members of historical Christian minorities, such as Armenians, enjoy relative freedom in Iran, the story is different for those who have converted to Christianity from Islam.
Muslim converts face intimidation, not only from their families, but also from Iranian authorities.
The Hungarian government learned of Strandsson's story and anounced on its official website that it was willing to help her.
The statement said: "The Hungarian State is ready to recognise the girl from Iran as a refugee who the Swedish authorities intend to send back to the Islamic Republic despite the fact that her life would be in considerable danger on account of having converted to Christianity."
Deputy Prime Minister Zsolt Semjén, who is currently performing the duties of Prime Minister as Viktor Orban is currently on leave, told a local newspaper: "If Aideen Strandsson seeks asylum in Hungary, at the end of the procedure - which naturally also involves national security screening - she may expect a favourable ruling as Hungary does not send anyone back to a country where his or her life or physical well-being may be in danger.
"Right from the beginning, we have differentiated between economic migrants and genuine asylum-seekers. We protect Hungary from the invasion of migrants, but we provide help for genuine refugees, those whose lives are in direct danger on account of their religion, nationality or political affiliation."
Her deportation would arguably be a violation of the Geneva Convention on Refugees which asserts that a refugee should not be returned to a country where they face serious threats to their life or freedom.
Strandsson, whose application for asylum was recently rejected by the Swedish authorities for the second time, arrived in Sweden in 2014, and soon thereafter converted to Christianity by asking for a public baptism.