The Brazilian full-back took Van Persie's jersey as he and the former Gunners captain left the pitch at half time during Saturday's Premier League defeat at Old Trafford.
Ray Parlour, who played almost 500 games for Arsenal between 1992 and 2004, commented: "That's a joke. I wouldn't have that. At half-time? He should be concentrating on his game and he was having a nightmare.
"He can't play in a back four as he gets caught out of position all the time."
TV presenter Piers Morgan tweeted: "If Santos really asked for, and got #VanPursestrings' shirt as they walked off at half-time, he should never play for Arsenal again."
Santos, 29, moved to the Emirates Stadium in a £6.8m deal from Fenerbahce in August 2011.