Forest value Johnson, 22, at £50million and have already rejected three bids from Brentford for the Wales international, the last believed to be worth £43m.
Spurs had proposed players in part-exchange for Johnson during talks and are poised to table a bid, amid hope that Forest's negotiating position could soften as tomorrow's 11pm transfer deadline approaches.
Spurs were in talks with Barcelona over a straight loan for Fati, 20, but the winger agreed to join Brighton for the rest of the season, and is set to undergo a medical on Thursday.
Atletico Madrid could revive their interest in Hojbjerg, who has also been offered to Manchester United, with a loan offer but Spurs would insist on any deal including a purchase obligation or option at the end of the campaign.
Gallagher would be allowed to leave Stamford Bridge for around £50m amid fierce competition for places in Chelsea's midfield, with Cole Palmer expected to be the Blues' final signing of the summer in a £45m deal from Manchester City.
Spurs are still working to offload Hugo Lloris, Tanguy Ndombele, Sergio Reguilon, Japhet Tanganga and Eric Dier before the deadline, while Davinson Sanchez is among the other players for whom they would consider offers.