Rossi, who scored four goals in 14 games for United between 2005 and 2007, has been invited by Solskjaer to rebuild his fitness and form at the club's Carrington training ground, according to BBC Sport's Simon Stone.
After United's 1-0 win at Tottenham Hotspur on Sunday, Solskjaer confirmed Rossi is aiming to rehabilitate himself and earn a move to a different club, rather than to sign on for any role at United.
"We're not going to sign Giuseppe, I don't think," the Norwegian said, according to NBC Sports' Joe Prince-Wright. "Bit of help to get him fit and if anybody wants to sign him, they should act quick."
Rossi got to train alongside and learn from Solskjaer during his time as a young striker at United, but he didn't enjoy any great success in England. The 31-year-old has endured an injury-ravaged career to date through spells at Villarreal, Fiorentina, and most recently Genoa.
At the end of last season, he failed a drug test following a Serie A match against Benevento, testing positive for the banned substance dorzolamide. Though a one-year ban from football was mooted, he was ultimately given only a warning and is free to continue his playing career.
Some reports had suggested Rossi may be brought onto Solskjaer's coaching staff, but the player himself appeared to emphatically rule that out.
Rossi, a former Italy international who won 30 caps, scored one goal in nine Serie A games for Genoa last season.