The move will be seen as a coup for the former champions, the sport's most successful team after Ferrari but well off the pace this year, who have more arrivals in the pipeline as they build up to a partnership with Honda in 2015.
"He's signed a contract with us," Whitmarsh said at the Japanese Grand Prix, without giving any details about a starting date.
"It is not a one-off thing we are doing, as we have made some moves during the course of this year. There are some other things we are doing to strengthen our team," McLaren managing director Jonathan Neale told reporters later.
"It is a series of moves as we head towards 2015 with Honda to make sure this team stays at the forefront of F1 and is set for winning ways."
Red Bull team boss Christian Horner indicated McLaren could expect a long wait before Prodromou, head of aerodynamics in Adrian Newey's design team, would be free to return to his former employers at Woking.
"The contents of any contract will be confidential between the company and the individual," Horner told Reuters.
"We've enjoyed a long relationship with Peter and he's a valuable member of the team and there's quite a duration left on his contract.
"We won't force him to stay, if it's his choice to leave then he'll leave at the end of his contract. But we are quite some way off that."
Horner, who could be celebrating Sebastian Vettel's fourth successive title for Red Bull on Sunday, would not confirm that Prodromou's current contract ran to 2015.
Vettel said Prodromou had played a big part in Red Bull's success.
"He's obviously been a very, very strong character in the background responsible for a lot of good stuff that has happened to us over the last couple of years," the German told reporters after qualifying.
"For sure I hope he stays with us but I don't know. I only just heard, when I got out of the car."
London-born Prodromou was McLaren's head of aerodynamics until 2006, having joined in 1991, when he followed Newey to Red Bull. Together, they have produced the cars that have won the last three drivers' and constructors' titles.
Whitmarsh had indicated at the Korean Grand Prix last weekend that some high profile technical signings were coming for a team without a podium finish this season.
"We've been recruiting in the technical team, strengthening that quite a lot. There have been quite a few new starters this year, there are some as yet unannounced that will be headline-grabbing when they get announced," he told reporters.
"We have to look at ourselves as a team, from the top management all the way through, as well as our technical team, to ensure we return to winning ways as quickly as possible."
Jenson Button, the 2009 world champion, who partners Mexican Sergio Perez in the current lineup, welcomed the positive news in a difficult season.
"The people working inside this team, and have been for many years, are still doing a mega job but it is great we are adding to their skills and experience," he told reporters.
"It is great that Jonathan, Martin and everyone else are looking outside the box, which is going to help us more in the future."
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