Arsenal star Sanchez was wanted by Manchester City while Liverpool were chasing Van Dijk before the 11pm deadline.
But White tweeted: “They stay! Sources telling us Alexis Sanchez stays @Arsenal while Virgil Van Dijk stays @SouthamptonFC!”
Late deals were beginning to trickle in but the story of transfer deadline day are the players who stayed put.
Sanchez is set to remain at Arsenal after the Gunners' bid for Monaco forward Thomas Lemar, for as much as £92million, stalled.
Lemar, 21, has been a target of Arsenal boss Arsene Wenger for much of the summer, with Liverpool also interested in him.
It is understood the Gunners made a late push to sign him after Manchester City made an improved offer for Chile forward Sanchez.
However, Express Sport understands the Gunners are now not expecting to get the deal over the line before the 11pm deadline, and ultimately means Sanchez is likely to stay at Arsenal.
Southampton defender Van Dijk is not expected to get his wish of a move away, with Saints holding firm in their stance that the Dutchman, a target for Liverpool, is not for sale.
Manchester City have ended their interest in West Brom defender Jonny Evans.
The future of Riyad Mahrez remains unclear. Mahrez, a key figure in Leicester's surprise title triumph in 2016, has been allowed to leave the Algeria squad and return to Europe to finalise his transfer to his new club.
Roma have had three bids rejected for the 26-year-old, the third for £32m, while Chelsea are reportedly still interested in the 2016 PFA Player of the Year.
The big move of the day so far is Alex Oxlade-Chamberlain's switch from Arsenal to Liverpool for £35m.
The 24-year-old England midfielder has agreed a long-term contract, understood to be five years, at Anfield.
Elsewhere, full-back Serge Aurier joined Tottenham from Paris St Germain in a £23m move.
Swansea landed Renato Sanches, the Portugal international midfielder, on a season-long loan from Bayern Munich and could re-sign Wilfried Bony from Manchester City after having a £12m bid accepted.
Bony appears likely to be a replacement for Fernando Llorente, the Spain striker who could join Aurier at Tottenham.
Llorente had been expected to be reunited with Antonio Conte - his former boss at Juventus - at Chelsea, but negotiations with Spurs are more advanced.
Crystal Palace are close to re-signing Liverpool defender Mamadou Sakho, who starred for them on loan last season, after they had a £26m bid accepted.
The Reds have let Divock Origi leave on loan for the season, with German club Wolfsburg his destination.
Mick Quinn PLEADS to van Dijk on behalf of Liverpool
As the evening progressed Everton made 19-year-old forward Nikola Vlasic their ninth summer signing, on a five-year contract from Hajduk Split.
Brighton signed defender Ezequiel Schelotto from Sporting Lisbon for an undisclosed fee, while Watford also raided the Portuguese club for full-back Marvin Zeegelaar and signed goalkeeper Orestis Karnezis and defender Molla Wague on loan from Udinese.
Arsenal striker Lucas Perez rejoined Spanish side Deportivo de La Coruna, a year after leaving, on a season-long loan.
Stoke duo Giannelli Imbula and Bojan Krkic have been loaned to Toulouse and Alaves respectively, and Nampalys Mendy departed Leicester for Nice on a season-long loan.
The light relief on a sometimes tense day was provided by West Ham striker Diafra Sakho, who returned from passing a medical at French club Rennes - via a trip to Chelmsford racecourse where his agent's horse won the 3.20 - to try to engineer his departure from the London Stadium.
West Ham are adamant the Senegal international is not for sale.
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