Hartlepool have announced they will not sign convicted rapist Ched Evans after boss Ronnie Moore expressed an interest.
Arsenal manager Arsene Wenger insists he will only spend money in January on the right player.
Leigertwood is available on a free transfer after seeing his contract with the Royals expire and he has been training with Stuart Pearce's side.
The 31-year-old was due to feature in last week's friendly game at Ilkeston but he suffered a minor knee injury during the warm-up. However, he will get the chance to impress this week when the Reds do some warm-weather training.
"He is on trial. It was unfortunate against Ilkeston, he just turned and twisted his knee a little bit during the warm-up," Forest assistant manager Steve Wigley told the Nottingham Post.
"But we said to him that there are two games in Portugal coming up and he will be with us. He has done really well in training. He is wise enough to know that it was not right to risk it. He will get his chance.
"We are looking to add to the squad. At one end of the spectrum we are looking at players who will cost us money. But at the other end, you are also looking for players who are still hungry and he falls into that category, you can see that.
"He has done well in training and the fact is that we have not got a midfield player of his type, in a powerful figure with real physical presence.
"He has been promoted with clubs before, he has played in the top flight as well. Sometimes football is funny. Sometimes you can pay a couple of million for a player, other times you can pick them up for nothing - and you still get the same thing.
"We are prepared to look at anyone who can come in and give us something that we haven't got. That is the test. We want to try to find something that is a little bit different."
Leigertwood has also played for Crystal Palace, Sheffield United and QPR.