The latest news on the health of the 53-year-old Glasgow-born presenter said he was making progress as he recovers from the condition which has left him hospitalised since Tuesday.
His wife Jackie Ashley also expressed her thanks to the thousands of people who have sent their best wishes to the high-profile political journalist. They range from BBC colleagues to Labour's leadership.
A BBC spokesman said: "Andrew Marr is responding to medication and continuing to progress. He is conscious and eating. He and his family are touched by all the warm wishes and support flooding in."
Nick Robinson, the BBC's current political editor, tweeted: "Excellent news that Andy Marr is conscious and continuing to make progress after his stroke. All thoughts with him, Jackie and his family."
Andrew Neil, who hosts the corporation's Daily Politics Show, added: "Good news. Fingers crossed."
The corporation has said it will continue to broadcast The Andrew Marr Show and Radio 4's Start The Week with guest presenters in his absence.
His place on The Andrew Marr Show this week will be taken by journalist James Landale.