After spending 10 years at the club Wayne Rooney was tipped to take the captain's armband, but Van Persie skippered the Dutch national side at the World Cup and he enjoys a good relationship with the new man in charge at Old Trafford.
Last week Van Gaal said he would take time to decide who should take the honour, and Robson appears to think the Dutchman has already made up his mind.
"Robin was his captain of Holland, so that is an obvious one," he said.
"I would be really surprised if he didn't give Robin it."
The close bond between Van Gaal and Van Persie was clear to see throughout the World Cup.
The pair spoke publicly about their admiration for each other and Van Persie raced over to the bench following his stunning opening goal against Spain to give his manager a big high five.
Van Gaal defied his critics to guide Holland to the semi-finals of the World Cup before they were knocked out by Argentina on penalties.
Robson believes Holland's run to the last four bodes well for United, who finished seventh last year under David Moyes.
"He has been a winner all his coaching career so hopefully he will install that back into the club," said Robson, who played 461 times during a 13-year spell at United.
"In the World Cup Holland were very good. Tactically he got them really well-drilled and I thought they were very unlucky. I thought they were better than Argentina, and they could have gone all the way."
Van Gaal has put his squad through tough training sessions since landing in Los Angeles, where United kick off their pre-season campaign on Wednesday against LA Galaxy.
The players in the 25-man squad, which includes new signings Anders Herrera and Luke Shaw, have been keen to impress their new manager, who was appointed as Moyes' successor in May.
The pain and disappointment of last year's dismal showing under the Scot will also be driving the players on, according to Robson.
"The fans and the club were disappointed last year, but nobody hurts more than players," said Robson, who won two league titles, one Cup Winners' Cup and three FA Cups during his time at Old Trafford.
"I know these players. They have won the title, the Champions League, and a lot of them were really disappointed about what happened so the lads that are still at the club will want to bounce back, they will want to prove to people that they are top players and they can win things."
Robson thinks United could fall short though unless they sign a centre-back.
Last season United conceded 43 goals. After losing Vidic, Rio Ferdinand and possibly Patrice Evra, Robson is concerned United lack experience in their back line.
"When you lose three class defenders like that you need to bring in a really top-drawer centre-half to the club. That would really give us a chance of winning the title," Robson said.
"You look back over the years and United have had Nemanja and Rio as a pairing, Brucey (Steve Bruce) and Pally (Gary Pallister).
"Then we had Jaap Stam. He was a really dominant and aggressive player and we need one of those to come in to the club."
Robson believes Bruno Martins Indi would have been the ideal signing for United.
The 22-year-old Holland centre-back joined Porto from Feyenoord on Wednesday after enjoying a successful World Cup under Van Gaal.
"He really impressed me in Brazil, especially against Argentina when they were up against it," Robson said.
"He had a really good tournament. He had a big presence on the pitch and we need someone like that."
- Sports & Recreation
- Louis van Gaal
- Nemanja Vidic