The 31-year-old, who left United as a youngster in 2007, is without a club and has been training at Carrington recently.
United aren't going to sign the 31-year-old, but Solskjaer says plenty of clubs should be vying for his signature.
"We are not going to sign Giuseppe, I don't think," he told Sky Sports. "He's training well with us, he looks sharp. He's looking for a job and a club, so he's probably going to stay another week with us unless someone takes him.
"He scored a fantastic goal in training and yesterday the gaffer turned around and said 'what about that son?' We're helping him get fit before his next move. If there are any takers, they should be quick."