The two Premier League sides reached an agreement on the transfer fee.
However, White has been on holiday and will return this week to finalize the switch.
The Gunners had a £40m bid for the England centre-back rejected, while an improved offer of £47m was also knocked back.
Brighton have also inserted a sell-on clause for White, who featured in all but two of league fixtures last season as they finished 16th.
The 23-year-old was then included in the Three Lions’ Euro 2020 campaign and played in the two warm-up games against Austria and Romania before the tournament.
White would become Mikel Arteta’s third summer signing, following the arrivals of left-back Nuno Tavares from Benfica and midfielder Albert Sombi Lokonga from Anderlecht.
