The 26-year-old has been given permission to travel for a medical after the two clubs agreed terms following United’s decision to switch their attention from Chelsea’s Marc Cucurella.
Cucurella was on United’s shortlist but Chelsea’s decision to play him in Wednesday’s EFL Cup win over AFC Wimbledon meant he could not join Erik ten Hag’s side and then play for a third club this season if his loan spell at Old Trafford was cut short in January, when Shaw and Malacia are expected to be available.
Reguilon has been deemed surplus to requirements by new Spurs head coach Ange Postecoglou after spending last year on loan at Atletico Madrid.
The Blues turned down a £40m bid for Gallagher from West Ham in July.
Tottenham are also pursuing a deal for Nottingham Forest’s Brennan Johnson with the Midlands club valuing him at around £45m.