The new Barca coach wants Wijnaldum as another option in midfield, and the club have agreed to push for a deal, although their priority is to finish moving on unwanted players first.
Koeman -- who quit as manager of the Netherlands to take the job at Camp Nou -- wants to reunite Wijnaldum and teammate Frenkie de Jong at club level, believing the pair have proved effective for the national team.
There are some concerns, though, over Wijnaldum's age -- at 29 -- and the age profile of Barcelona's squad.
Wijnaldum joined Liverpool from Newcastle in 2016 and has a year left on his contract at Anfield.
He has been a first-team regular throughout his four seasons there, making 37 Premier League appearances in last season's title-winning campaign.
Koeman believes Wijnaldum will provide him with more possibilities in his preferred 4-2-3-1 formation.
However, the sale of Ivan Rakitic -- who joined Sevilla for a €1.5m fee plus up to €9m in variables on Tuesday -- and the imminent departure of Arturo Vidal mean the coach feels there is room for another, experienced alternative in the squad.
Sources have told ESPN that Vidal, 33, is now in advanced talks with Inter.
He will leave for a similarly low fee to Rakitic, but his sale will also free up his substantial wage packet, which could go towards financing the Wijnaldum deal.