The 31-year-old was a member of City's Championship winning side in 2004 but is without a team after being released by Luton last month.
Canaries boss Bryan Gunn has described the move as an "open ended" arrangement, but admitted that if McVeigh impresses there may be a more permanent contract available.
"He will be out there with the lads and he's been involved in the fitness testing in the last couple of days as well," Gunn told the Norwich Evening News.
"It's open ended. Paul and I had a chat during the summer and he told me about his contractual situation. I said 'Come in and join us for pre-season'. There are no strings attached either way but if he impresses not only me, but the other coaches, who knows?
"It gives Paul an opportunity to keep his fitness going. I'm sure he'll be speaking to other clubs and other managers and he's probably got an agent working for him as well. He's living locally so it does no harm and it helps us in terms of numbers, in terms of the training."