Manchester United had been heavily linked with Anderson before pulling out of the bidding race, having judged the asking price too high. The 23-year-old has since impressed at the World Cup, forming a strong midfield partnership alongside Arsenal's Rice for England.
Maguire, who returned to the England squad in March after a year out of the setup, said he had been struck by how good Anderson looked in training.
"I've heard there are many clubs abroad after him as well, but I must say, I was not in the England squad for 12 months, and I was interested to go in in March and just see there were 10, 15 new players who I hadn't played with," Maguire said.
"And I knew he was a good player, and in training, he impressed me even more than I knew. He's got absolutely everything. He reminds me of Declan Rice in some aspects in terms of his profile."
Maguire went on to praise the range of attributes Anderson brings to the team.
"He's good on the ball; he can drive with the ball. He's got an eye for goal as well, to say he's playing number six at the minute, but he's one of those, I think he could play 10, 8, 6. He's that good," Maguire said.
Anderson left Newcastle United for Nottingham Forest before establishing himself as one of the Premier League's most coveted young midfielders, with Manchester City now leading the chase for his signature this summer after United's withdrawal from the race.
His World Cup form alongside Rice has only strengthened his market value, with Maguire's praise from inside the camp adding further weight to a reputation that continues to grow with every England appearance.