McNamara is understood to have been hospitalised after collapsing near his home in York on Saturday.
"Thank you all for your kind words and support," the 46-year-old's daughter wrote on his Twitter account on Monday evening.
"My dad is getting the best care possible, as you all can understand this is difficult for us as a family and request privacy at this time."
This is Jackie’s daughter.
Thank you all for your kind words and support. My dad is getting the best care possible, as you all can understand this is difficult for us as a family and request privacy at this time. ❤️
— Jackie McNamara (@Jackie_Mc4)
McNamara spent 10 years at Celtic between 1995-2005 after beginning his senior career at Dunfermline, winning four Scottish Premiership titles and captaining the team before joining Wolves.
He also represented Aberdeen, Falkirk and Partick Thistle before retiring in 2011.
McNamara - who earned 33 senior international caps for Scotland - later managed Partick, Dundee United and York City and has been working as a players' agent in addition to advising Dunfermline.
"The thoughts and prayers of everyone at #CelticFC are with Jackie and his family," Celtic wrote on Twitter.
The official account of the Scotland national team said: "The thoughts of everyone at the Scottish FA are with former Scotland player Jackie McNamara and his family."