The England international currently plays for Atletico Madrid and could arrive in north London for a reported fee of around £34.7 million.
Tottenham have long admired the 25-year-old, who has made 77 appearances in 18 months with the Spanish club and scored seven goals.
However, Gallagher has struggled to secure a regular starting spot under Diego Simeone.
According to reporting by Press Association, his signing would help address Tottenham's injury woes in midfield.
Rodrigo Bentancur is expected to be out of action for at least three months after being forced off during the closing stages of Wednesday’s 3-2 defeat at Bournemouth.
Head coach Thomas Frank described the injury as "a bigger one" ahead of Tottenham’s FA Cup loss to Aston Villa, with the midfielder now set to undergo hamstring surgery.
Bentancur’s absence, alongside Lucas Bergvall, left Spurs without a recognised midfielder on the bench for the Villa match.
Pape Sarr and Yves Bissouma were also unavailable due to Africa Cup of Nations duties.
The potential arrival of Gallagher comes at a critical time for Tottenham, who currently sit 14th in the league and are without several key players due to injury and international commitments.
His addition would also be timely ahead of Saturday's home fixture against West Ham United, where fan group Change for Tottenham has organised a protest.
Gallagher was previously linked with a move to Spurs in the summer of 2024, but the deal did not materialise.