The Danish defender impressed in his season on loan with Fulham last season, taking the captain's armband despite not being a permanent member of Scott Parker's squad.
Palace are keen to invest in their squad this summer and bring the average age down, with Marc Guehi (21) having already joined in a deal which could reach £20million and Michael Olise (19) having been added from Reading for £8million.
Standard Sport understands Andersen, who was also a target for London rivals Tottenham this summer, will cost an initial €22million, with €5million in add-ons to follow.
The 25-year-old, who is expected to sign a five-year deal and underwent his medical in Greece on Friday, was part of the Denmark squad which reached the semi-finals at Euro 2020, where they lost to England at Wembley.
Palace could still move to keep experienced defender Gary Cahill, whose deal at Selhurst Park expired at the end of last season. Negotiations have been ongoing over a new contract, but the two parties are apart on wages with the Eagles looking to cut the former Chelsea defender's basic pay. There is also interest in Schalke defender Ozan Kabak, who spent time on loan at Liverpool last season.
Andros Townsend, Scott Dann and Mamadou Sakho, among a number of others who were out of contract, have been allowed to leave the club already.
Patrick van Aanholt is another who has left and has joined Galatasaray, meaning Vieira will likely want to bring in cover and competition for Tyrick Mitchell at left-back.
The Eagles are also looking to bolster midfield. Olise has arrived but will miss the start of the season through injury. Palace are competing with Leeds United over a loan move for Chelsea's Conor Gallagher - the midfielder they had all but signed on loan last summer only to pull the plug and add Michy Batshuayi instead.