The report claims both United and City have scouted the Dutchman on several occasions already this season, and have left impressed each time.
The 22-year-old Celtic star cost the Scottish side just £2.6m when he moved from FC Groningen in June 2013.
And United and City will be expected to pay at least £10m if they want to sign him this summer. Arsenal officials have also watched van Dijk this season, but United and City are thought to be leading the race.
United are set to lose captain Nemanja Vidic at the end of the current campaign, and Patrice Evra may also follow the Serbian out when his contract expires.
Replacements will be required and United boss David Moyes has been linked with a number of defenders as he's expected to strengthen.
The defending Premier League champions have been uncharacteristically leaky at the back this season, and with money to spend, Moyes is likely to target additions in central defence.
Similarly, Manchester City boss Manuel Pellegrini wants to add competition at the heart of his defence, with only Vincent Kompany guaranteed a place.
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