Spurs agreed a fee of £47.5 million ($59.8m) fee plus add-ons with Forest, sources told ESPN, with Johnson signing a six-year deal.
"(Johnson's) departure brings an end to his 14-year association with the club, having joined the Forest Academy at the age of eight," Forest said in a statement.
Johnson made 108 appearances for Forest, scoring 29 goals, and starred in their promotion to the Premier League in 2022.
He has 20 caps for Wales and was part of their squad for last year's World Cup.
However, Tottenham appear to have failed in their deadline-day pursuits of Chelsea's Conor Gallagher and Bournemouth's Lloyd Kelly.
The midfielder has started all four matches this season under Mauricio Pochettino and was named captain in Wednesday's EFL Cup second-round win over AFC Wimbledon. Sources have told ESPN that Gallagher, who has two years remaining on his contract at Stamford Bridge, is valued in the region of £50m.
Spurs were also looking to strengthen at centre-back after having a £20m bid rejected by Bournemouth for Kelly.
Information from Reuters was used in this report.