Earlier on Friday, Spurs made a verbal offer of £40million for Gallagher, whose Chelsea future had been thrown into doubt by the Blues' capture of Cole Palmer from Manchester City, but the deal was dependent on Hojbjerg finding a new club.
The Dane turned down interest from Fulham and Spurs could not agree a deal with Atletico Madrid, who were exploring a last-minute loan move for the midfielder. Spurs wanted to include an obligation to buy Hojbjerg, who has two years remaining on his contract, but Atleti could not come up with a suitable package.
Hojbjerg is now set to stay at Spurs, and is expected to return to the bench for Saturday's Premier League game at Burnley.
Spurs also saw a deadline-day bid, believed to be in the region of £20m, for Bournemouth's Kelly rejected but did not pursue their interest after failing to move on Dier. Davinson Sanchez's future was still in balance with the 11pm BST deadline looming, amid interest from PSV Eindhoven.
Dier was suggested as a makeweight to Fulham and Bournemouth in separate deals but would have only left Spurs for a big move this summer, and is now expected to run down his contract and leave on a free in a year's time.
Harvey White is still hoping to leave before the deadline, while there is an increasing expectation that Spurs could cancel the final year of Hugo Lloris' contract after failing to find a buyer for their former captain.