Ian Paisley's family are maintaining a bedside vigil as the 85-year-old remains in intensive care, according to reports on Tuesday.
The former Northern Ireland First Minister is in hospital in Ulster after being admitted suffering from heart problems.
A statement was issued on behalf of his wife Baroness Paisley saying: "She requests that the family's privacy be respected at this difficult time."
Paisley, once a fierce opponent of sharing government powers with nationalists and republicans in Northern Ireland, was elected First Minister in May 2007 with Martin McGuinness, a former IRA leader in Londonderry, as Deputy First Minister.
It was a remarkable partnership, the two men becoming firm professional and personal friends, and who were later nicknamed the "Chuckle Brothers".
Paisley underwent tests for an undisclosed illness in summer 2004 and afterwards admitted he had "walked in death's shadow". Some years later he had a pacemaker fitted after feeling unwell at the House of Lords.
His son told the BBC on Tuesday he had spent the night at hospital with his father but did not go into details about his condition.
Paisley was a fierce critic of the 1998 Good Friday Agreement which led to the formation of the first power-sharing administration at Stormont since 1974.
But in the aftermath of the signing of another political arrangement which became known as the St Andrews Agreement of 2006, he underwent an astonishing political transformation which culminated with him going into power with Sinn Fein a year later.
It was a deal which would have been unimaginable at the height of the IRA terrorist campaign, but this was a much different Ian Paisley from the firebrand preacher who spent decades on the margins of political power, damning the Catholic Church, and who was once thrown out of the European Parliament for denouncing Pope John Paul II as the anti-Christ.
He stood down as First Minister in May 2008 with his long-time deputy party leader Peter Robinson taking over around the same time as he was made a life peer in Gordon Brown's Dissolution Honours List. He was MP for North Antrim from June 1970 until May 2010.