The former Manchester United and Ajax goalkeeper, Edwin Van der Sar was taken to an intensive care unit in Croatia on Friday, where he was on holiday.
"Edwin is still in the intensive care unit, but is stable," Van Kesteren said in a statement issued by Ajax.
"Every time we get to visit him, he's communicative," she added.
"We have to wait patiently to see how his situation will develop. He is not in life-threatening danger."
The former Netherlands international, who won 130 caps for his country, resigned from his role as Ajax chief executive in May after the side finished third in the Dutch league and failed to qualify for the Champions League for the first time since 2009.
He retired from playing after leaving United in 2011 before joining the Ajax board in 2012, and later becoming the club's chief executive in 2016.
Edwin Van der Sar made 266 appearances for the Red Devils and helped them to win four Premier League titles and the 2008 Champions League.
Van der Sar, 52, had been on holiday in Croatia when he reportedly suffered a cerebral haemorrhage.
Manchester United sent a message of support for their former goalkeeper, tweeting: "Sending all our love and strength to you, Edwin."
Fulham, with whom Van der Sar spent four years before joining United, tweeted: "Everyone at Fulham Football Club wishes Edwin a speedy recovery. We're thinking of you."
The Professional Footballers' Association tweeted: "The thoughts of everyone at the PFA are with Edwin and his family."
Edwin Van der Sar won the Premier League four times during his six years at United, as well as the Champions League in 2008, which he also lifted while at Ajax.
After retirement, he returned to Ajax where he took up the role of chief executive in 2016 but left the post at the end of last season.
Following a disappointing campaign, the club finishing third in the Eredivise and missing out on Champions League qualification, Van der Sar announced he would be stepping down from his role after almost 11 years on the club's board.