Barca's previous offer to Araujo was worth around €5 million annually, with sources saying the money on offer from the Premier League sides was worth almost double.
However, Barca have now upped their proposal, with the player's agent, Flavio Perchman, revealing an agreement is close after a meeting with sporting director Mateu Alemany on Tuesday.
"It's close," Perchman told reporters. "There is a desire on both sides [to finalise the renewal] and Ronald wants to stay at the club."
Araujo, 23, has been on the same deal at Barca since 2018, when he first joined the "B team" from Boston River in his homeland in a deal worth an initial €2m.
That made him one of the lowest-paid players in the first team going into the current campaign, with his agents keen for his salary to rise along with his status at the club.
Under coach Xavi Hernandez he has emerged as a key player this season, making 34 appearances in all competitions and scoring four goals, including in the recent 4-0 Clasico win over Real Madrid at the Santiago Bernabeu.
With Barca having to adhere to LaLiga's strict financial rules due to their financial problems, though, they have played hardball during negotiations with Araujo's camp.
ESPN previously reported that Christensen will join the club from Chelsea on a free transfer in the summer, but a move for Rudiger is less likely.
The public nature of those negotiations forced the club to up their offer to Araujo, who was willing to accept a lower deal to remain with the LaLiga side.