The Dutchman quit Southampton on Tuesday and was swiftly named as Roberto Martinez's successor at Everton following something of a saga.
The South Coast club reportedly picked up £5 million in compensation from the Toffees (The Telegraph).
Koeman had only spent two years at St Mary's, having replaced Mauricio Pochettino in 2014, but the 53-year-old had arguably did a better job than his Argentine predecessor.
He brought in better players and as a result Southampton qualified for the Europa League preliminary rounds in both of Koeman's seasons.
Pochettino - now at Tottenham Hotspur - had succeeded the axed Nigel Adkins, who had guided the club back into the Premier League from League One, and Fonte has reminded concerned parties that Southampton know about this process.
"It's not ideal, but we've been through situations like this before," he told The Mail.
"We have to trust the board and the club that we are going to get it right and appoint someone with quality. We are a top club in England now.
"I'm sure the club will sort this situation out like in previous occasions. We've been through this before. We know how to deal with it at least."