The royal couple toured the new ?105 million ($170 million, 130 million euro) St George's Park complex at Burton-upon-Trent, which will be the training base for all 24 England teams from junior to senior levels.
William and Catherine met members of the senior squad, who are training at the facility in central England ahead of the World Cup qualifiers against San Marino and Poland.
They were introduced to players including captain Steven Gerrard and Frank Lampard, and there was laughter from the squad as William shook Cole's hand.
Cole's place in the team for Friday is uncertain after he made a foul-mouthed Twitter outburst at the Football Association, English football's governing body, following accusations he had provided "evolving" evidence supporting Chelsea teammate John Terry in his FA racism hearing.
The Times reported that William had raised the subject, telling Cole: "If you continue to be a naughty boy, they will take your Twitter account off you."
Cole was seen laughing and then held his head in his hands in mock shame.
The prince, who is a fan of English Premier League side Aston Villa and also the president of the FA, praised the "wonderful facilities and beautiful surroundings" at the centre.
"St George's Park, and the concept that underpins it, is something totally new. It will be far more than just a world-class facility for training our future world-beating national team," he said in a speech.
"It is more than just the university from which thousands of highly qualified coaches will graduate.
"It is also a magnificent example of the sort of social initiative that brings opportunity and purpose to wider British life."
The prince could not resist a dig at his brother, who is serving as a helicopter pilot in Afghanistan.
"I feel tempted to cry 'God for Harry, England and St George' (from William Shakespeare's play, Henry V) but I really don't want to lower the tone by bringing my brother into it," he said.
Catherine, who was caught up in her own furore last month after several European media outlets published photographs of her sunbathing topless, wore a grey coat by Reiss and black boots.
Source: AFP
Source: AFP