The 26-year-old has made only two appearances for United this season after being sent out on loan to Fiorentina in the previous campaign.
Anderson has spoken of his eagerness to go back to Brazil and would relish the opportunity to spend a second spell with Gremio.
He told Radio Bandeirantes: "There is always the possibility of returning to Gremio. It’s my home, I grew up at the club. I am a fan, everyone knows that.
"It's possible, as it was in the past. I have been contacted by the club before.
"I want to be closer to my family, to my daughters. There has never been an offer, but I have known the club for a long time.
"I am not desperate to join Gremio, though, as I respect all Brazilian clubs.
"Gremio have a chance but it doesn't depend on me, it depends on their board. I was close a year ago, before joining Fiorentina. I would accept a reduction in my wages."
Anderson left Gremio to join Porto in 2006, then moved to Old Trafford the following year.
