The Brazilian swapped shirts with the Dutchman at half-time during the two teams' clash at Old Trafford in the Premier League last month.
The gesture infuriated some Arsenal fans and Santos eventually publicly apologised to his team-mates and the supporters.
However, the 29-year-old has now played down the incident and says that he is good friends with Van Persie.
"We Brazilians are used to it," Santos told Globo Esporte. "For me, it will not change anything.
"Van Persie is a great friend. We talk weekly, he is an extraordinary guy.
"Whenever he sees a bad expression on my BBM (instant messenger), he supports me, speaks positive words."
Santos also played down talk of a departure from Arsenal despite his limited first-team appearances so far this season.
"I received calls from interested clubs in Brazil, Turkey and Italy, but it's just speculation," he said. "Nobody handed me anything. The coach, who's also a manager, never said anything to me.
"I know he is very transparent and I am focused on staying at Arsenal, winning a title, doing well in the Champions League, and I have three more years of contract.
"I am happy here. I hope to play more and have more opportunities."