Spurs went into deadline day with the hope of adding more players than just Nottingham Forest’s Brennan Johnson, who is set to join the club in the coming hours.
Gallagher has continued to be on their radar all summer and Spurs also tabled a bid to sign Kelly from Bournemouth.
However, Tottenham have failed to move on the players they would have needed to offload to sign those two players.
It has been claimed that Spurs are no longer trying to sign either Gallagher or Kelly in the final hours of the transfer window.
Bournemouth have been wary about losing Kelly, a versatile defender, and were reluctant to do a deal late in the window.
Gallagher has always been keen to stay at Chelsea and has found a supporter in new manager Mauricio Pochettino.
Johnson is set to complete his move to Spurs and his arrival will mark the end of their incoming business for this summer.