Shagufta Kauser and her disabled husband Shafqat Emmanuel (pictured above) were sentenced to death in 2014 for allegedly sending blasphemous text messages in English.
They were thought to have sent them to a local Imam and other Muslims, according to the British Pakistani Christian Association.
Mr Emmanuel said at the time that he was tortured, forcing him to confess to the crime.
Lord Alton raised this previously with Lord Ahmad of Wimbledon, the government's special envoy for freedom of religion and belief, in May and was told that the government is monitoring the case.
This week David Alton, a crossbench peer, returned to the issue asking Baroness Sugg, the Parliamentary Under-Secretary of State for International Development, how much aid the UK was sending to Pakistan and whether that was being used to support persecuted minorities at all.
Baroness Sugg replied: "My Lords, in the past 10 years, the UK has given £2.6 billion in aid to Pakistan, targeted towards the poorest and most excluded, who are often from minorities. We promote minority rights from grass roots to the highest levels of government. UK aid to Pakistan is declining but continues to focus on the poorest.
"Since 2011, UK aid has supported primary education for 10 million children, skills training for almost 250,000 people, and microfinance loans for 6.6 million people."
Lord Alton thanked her and then asked: "Is she able to intervene on behalf of Shagufta Kauser, an illiterate woman from one of Pakistan's beleaguered minorities, who now occupies Asia Bibi's cell in Multan and who, like her, has been sentenced to death for allegedly sending blasphemous texts in English?
He went on to describe a barrage of atrocities happening in Pakistan: "When two children are forced to watch a lynch mob of 1,200 burn alive their parents; when no one is brought to justice for the murder of Shahbaz Bhatti, Pakistan's Minister for Minorities; when 1,000 Hindu and Christian girls are forcibly married and converted; and when minorities are ghettoised into squalid colonies, which I have visited, and forced to clean latrines and sweep streets, is it not time that DfID re-examined its policy of refusing to specifically direct any of the £383,000 that, on average, we give every single day to Pakistan in aid for the alleviation of the suffering and destitution of these desperate minorities?"
Baroness Sugg replied that she was concerned by the ongoing issue of blasphemy laws in the country and that they regularly raise them with the Pakistani government.
She didn't reference Shagufta Kauser specifically but said: "We are working to ensure that we understand where our aid is going. I can reassure the noble Lord that we continually keep our programmes under review, and where we can better prioritise resources we will do so."
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