Since Joan Laporta took over as Barca president, the club has been trying to reduce their wage bill in order to keep Messi and stay within La Liga's strict financial control rules.
La Liga chief Javier Tebas said last week that Barcelona, which has a total debt of more than 1 billion euros (US dollar 1.18 billion dollars), would not be shown any leniency.
Messi's last contract, signed in 2017, was the most lucrative in world sport according to a January report in newspaper El Mundo.
The club have been trying to rebuild the squad with Junior Firpo, Jean-Clair Todibo and Carles Alena being sold to make way for free signings Sergio Aguero, Memphis Depay and Eric Garcia.
Messi, 34, won his first major international title with Argentina over the weekend when they beat rivals Brazil in the Copa America final.
Messi was elected the tournament's joint best player along with Neymar after finishing as the tournament's joint top goal scorer with four goals while he also topped the assists charts with five.