United are currently without a captain following the departure of Nemanja Vidic to Inter Milan this summer, while Patrice Evra is close to joining Juventus.
As it stands, Wayne Rooney and Van Persie are the two leading candidates, but Robson believes that it will be the latter that gains the role thanks to his national team link with incoming boss Louis van Gaal.
"Robin was his captain of Holland, so that is an obvious one," Robson told the Press Association.
"I would be really surprised if he didn't give Robin it."
United are currently in America as part of their pre-season.