Van Persie was the skipper of the Netherlands national team for this summer’s World Cup in Brazil and is among the favourites to take the armband for the upcoming season at Old Trafford.
Nemanja Vidic and Patrice Evra have been captaining United for the last few seasons, but with the former having departed for Inter Milan and the latter reportedly on his way to Juventus, a new appointment will be required.
"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."
Robson, who made over 400 appearances for United between 1981 and 1994, was well known for his dogged style of play and believes that the Red Devils also need a commanding centre-half.
Only Chris Smalling, Jonny Evans and Phil Jones now remain as central defensive options, following the departure of Vidic and Rio Ferdinand.
"When I look at us, I just think losing Nemanja and Rio and conceding goals last year means can we get a really good centre-half? I think that's more important than a central midfield player at this moment in time,” Robson added in The Mirror.
"When you look back, we've nearly always had a really dominant player who is brilliant in the air and an organiser. If you go back, we've had Rio, Nemanja, Steve Bruce, Gary Pallister and Jaap Stam.”
United have so far recruited left-back Luke Shaw and Spanish midfielder Ander Herrera for a combined £60million, but must surely be on the prowl for a centre-back.
Aston Villa’s Ron Vlaar and Arsenal’s Thomas Vermaelen have been linked with moves to Old Trafford, but nothing concrete has yet transpired. Van Gaal could do worse than listening to one of United’s greatest ever players with regard to his transfer plans.