The 23-year-old's future is currently the subject of much speculation, with West Ham United believed to be keen to sign him during the summer market.
However, according to Football Insider, Spurs are also in the hunt for the England international, with the club's new head coach Ange Postecoglu viewing the midfielder as a priority target.
The report claims that Chelsea are open to selling Gallagher but will ask for around £40m for a player who featured on 45 occasions for the Blues during the 2022-23 campaign.
The Englishman had successful loan spells at West Bromwich Albion and Crystal Palace in the Premier League before becoming a first-team regular at Stamford Bridge.
Gallagher has another two years left to run on his contract at Chelsea, placing the club in a strong position when it comes to negotiating a price.
The midfielder featured on 35 occasions in the Premier League last season, but he only managed three goals and one assist during a disappointing campaign.
Chelsea are bidding to bring in Moises Caicedo from Brighton & Hove Albion this summer to boost a midfield department which has lost Mateo Kovacic, N'Golo Kante, Denis Zakaria, Ruben Loftus-Cheek and Mason Mount.
Gallagher was on the scoresheet in Chelsea's 5-0 win over Wrexham on Wednesday evening, and Blues boss Mauricio Pochettino is thought to be open to keeping hold of him in the current market.
The midfielder came through the youth system at Chelsea, representing their various age groups between 2008 and 2019 before debuting for the first team.
James Maddison has been Tottenham's biggest signing thus far this summer, with the 26-year-old arriving from Leicester City following the club's relegation to the Championship.
Harry Winks and Lucas Moura have been the club's two first-team departures in the current window, but there remains widespread speculation surrounding the future of Harry Kane.
Bayern Munich are still pushing to sign the England captain, with the Bundesliga giants emerging as the clear favourites to prise him away from the Tottenham Hotspur Stadium.
Spurs started their preparations for the 2023-24 campaign with a 3-2 defeat to West Ham, and Postecoglu's side will be back in action against Leicester on Sunday.