Pastor Yousef Nadarkhani has won the admiration of Christians around the world for refusing to recant his faith in appeals hearings – a condition for his acquittal.
He remains in Rasht prison in northern Iran, where he has been held since first being charged with apostasy while trying to register his church last year.
Mr Hague said, "I deplore reports that Pastor Yousef Nadarkhani, an Iranian Church leader, could be executed imminently after refusing an order by the Supreme Court of Iran to recant his faith.
"This demonstrates the Iranian regime's continued unwillingness to abide by its constitutional and international obligations to respect religious freedom.
"I pay tribute to the courage shown by Pastor Nadarkhani who has no case to answer and call on the Iranian authorities to overturn his sentence."
Pastor Nadarkhani was sentenced to death for "apostasy from Islam" last November, despite there being no such crime under Iran's penal code.
The Supreme Court upheld the sentence on the grounds that although the country's penal code did not criminalise apostasy, it was recognised as a crime by Sharia law and the Islamic Republic's spiritual leader, Ayatollah Khomeini.
Human Rights Watch added its voice to calls for Pastor Nadarkhani's immediate release. The New York-based group said the Iranian authorities should immediately free and drop all charges against him.
"Iran is one of the very few countries in the 21st century where authorities would drag an individual before a court of law and force him to choose between his faith and his life," said Joe Stork, deputy Middle East director at Human Rights Watch.
"Nadarkhani should not have to spend one more day in jail, let alone face execution."
The group voiced concern over the numerous arrests of evangelical Christians and converts to the faith in recent months. Although Iran has traditionally recognised Christian minorities like Armenians, Assyrians and Chaldeans, HRW said that officials were targeting evangelical and Protestant groups, which conduct their services in Persian.
Authorities accuse them of spreading religious literature in Persian in an attempt to attract Muslims to their faith.
Earlier in the year, the governor of Tehran province, Morteza Tamadon, described Christians as "deviant" and likened them to the Taliban, as he warned of numerous arrests in the near future.
Mr Stork called on Iran to meet its human rights obligations.
"Both international and Iranian law require Iranian officials to safeguard the equality and human rights of all Christians, regardless of whether they are historic communities such as the Armenians, Assyrians, and Chaldeans, or Christian converts," he said.
A written verdict is due any day on Pastor Nadarkhani's case. According to Christian Solidarity Worldwide (CSW), his lawyer Mohammad Ali Dadkhah denies receiving a verbal notification of the annulment of the charges from the court.
"Continued international vigilance and pressure is vital. The life of this man is still very much in the balance," said Mr Dadkhah.
CSW has been campaigning for Pastor Nadarkhani's release and facilitated in the sending of more than 19,000 emails from supporters to the Iranian embassy in the UK.
The group called on the international community to continue pressuring Iran on the pastor's release.
Chief Executive of CSW, Mervyn Thomas, said: "Until a written verdict is confirmed to have been received by credible sources, there must be no let up in pressure on the Iranian regime."