Both London clubs have been linked with a move for Cancelo, with Chelsea manager Antonio Conte looking to bolster his defensive options and provide competition for Marcos Alonso and Victor Moses.
Meanwhile Mauricio Pochettino is yet to replace Kyle Walker, who joined Manchester City for £50m (€54.9m, $64.2m) earlier this summer, and has lost Kieran Trippier before the season even kicked off, after the right-back suffered an injury in a friendly against Juventus.
However, Chelsea and Spurs will have to look elsewhere as the 23-year-old appears set to move to the San Siro on a season-long loan with option to buy, with Inter midfielder Geoffrey Kondogbia going in the opposite direction.
Italian newspaper Gazzetta dello Sport said on Tuesday (15 August), that Inter were eager to resolve Kondogbia's future as quickly as possible after the Frenchman refused to take part in training in a bid to force a move to Spain and was suspended indefinitely by the club.
Cancelo, who can also be deployed as right midfielder and wing-back, had attracted interest from Chelsea and Tottenham following a number of impressive performances for Valencia last season.
The Portuguese, who made 35 league appearances last term, spent a season at the Mestalla on loan from Benfica before making his move permanent in the summer of 2015 for £4.5m.
With Cancelo moving to Inter Milan, Chelsea have reportedly added Southampton defender Cedric Soares to their wish list, along with Juventus defender Alex Sandro and Paris Saint-Germain right-back Serge Aurier.
Soares has three years left on his contract at Southampton but he would be cheaper compared to other options, with Chelsea hopeful of having a bid of around £20m accepted for the Portugal international.
Aurier, meanwhile, has also been on the radar of Spurs and Manchester United and the former have reportedly reached a verbal agreement with the PSG defender, according to reports by French daily L'Equipe.
The Ivory Coast international looks set to leave Paris in the summer following the arrival of former Barcelona and Juventus right-back Dani Alves, but a potential move to the Premier League has been complicated by the uncertainty over whether he would be given permission to enter the UK amid an ongoing legal issue.
Aurier was given a suspended two-year prison sentence for attacking a police officer in 2016, an incident which saw him barred from entering the UK last November when his visa application revoked by the Home Office ahead of a Champions League tie with Arsenal.