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Sources have told ESPN that talks are at an advanced stage over a two-year contract and Hart could undergo a medical within the next 48 hours.
Tottenham are mindful of improving their homegrown quota to increase the overall squad available to head coach Jose Mourinho.
Premier League rules dictate a maximum of 17 overseas players can be named in any 25-man squad.
Mourinho only selected four homegrown players in February which restricted him to a maximum overall squad of 21.
Spurs have balanced the situation somewhat with the departures of Dutch goalkeeper Michel Vorm and Belgian defender Jan Vertonghen but sources have reiterated to ESPN that the club are also looking to extend Lloris' contract. The Frenchman's deal expires in 2022.
Hart is still expected to join regardless as he seeks to revive his career after making just three appearances for Burnley last season, all of which came in cup competitions. His last League appearance came in Burnley's 5-1 home defeat to Everton on Dec. 26, 2018.