While that initial offer was rebuffed, and discussions involving Alexis Sanchez in a potential cash-plus-player deal were short-lived, Barcelona is now confident of wrapping up the signing in a straight cash deal.
The Spanish giant is expected to table an improved offer that will meet the release fee stipulated in the contract Suarez signed in December, with the figure just over the 88 million euro mark.
Liverpool is still pursuing Sanchez but it is behind Arsenal in the race to land the Chile World Cup star, who is not a part of the deal to take Suarez to Camp Nou.
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The Merseysiders now expect to receive the third-highest transfer fee for a player at a Premier League club, beaten only by Gareth Bale's 100 million euro move to Real Madrid last summer and Cristiano Ronaldo's 94 million euro transfer from Manchester United to the Bernabeu in 2009.
Liverpool's American owners vowed to stand firm on Suarez's valuation after he had tarnished his image by biting Giorgio Chiellini and incurring a four-month ban from FIFA. Fenway Sports Group was adamant during talks with Barca that Suarez could only leave if the valuation of his release fee was met.
All parties now expect the 27-year-old to transfer to Camp Nou and line up alongside Lionel Messi and Neymar in a star-studded front line next season.
Barca went public with its interest in the Uruguayan on Tuesday when sporting director Andoni Zubizarreta praised "humble" Suarez for apologizing via Twitter for biting Chiellini.
This open display of affection for the player came just 24 hours before a meeting in London between Barca and Liverpool, which had no involvement in Suarez's apology.
"Everyone knows that Luis is an excellent player, everyone knows last season he played in various positions in attack, but everyone also knows that he is a Liverpool player," Zubizarreta said. "Regardless of whether the club signs him or not, his quality is unquestionable. He has shown a lot of quality with Uruguay and scored more than 30 goals with Liverpool, so he has the necessary quality to play for Barca."