Barcelona appear to be leading the race among clubs linked with the 27-year-old Uruguayan, who was expelled from the World Cup and handed a nine-match competitive international ban as well as a four-month sanction from any football activity for biting Italy defender Giorgio Chiellini. (Agencies)
A senior Liverpool source said that Wednesday's talks had been productive and would continue, according to reports.
"There are sensible expectations on both sides. Further talks and discussions will take place but no deal has yet been finalised," the source added.
Sky Sports television reported a similar outcome.
The meeting between the Merseyside club officials and a delegation led by Raul Sanllehi, Barcelona's director of football management, had been scheduled to discuss the future of last season's Premier League top scorer.
In order to prise Suarez away from Liverpool, the Spanish club will either need to meet a release clause, reportedly around 70-80 million pounds (USD118-USD134 million), in his revised contract or possibly offer a player-plus-cash deal.
Should the Merseysiders opt for the latter agreement, Barcelona's Chilean striker Alexis Sanchez - a Liverpool target in the past - appears to be the Anfield side's preferred choice as Suarez's replacement.
However, club president Josep Bartomeu said on Wednesday that Barcelona were "counting on" Sanchez for next season despite enquiries from many clubs.
After initially denying that he bit Chiellini in Uruguay's 1-0 win in Natal on June 24, Suarez issued a carefully worded statement on Monday, admitting his offence and vowing there would be no repeat.
Barcelona sporting director Andoni Zubizarreta stoked rumours of a possible move for Suarez to the Catalan club on Tuesday by praising him for his 'humility' in admitting his mistake.
In the run-up to the tournament in Brazil, Suarez scored 11 goals in 16 qualifying matches to help Uruguay secure a berth at the finals, while at Liverpool, his 31 league goals was the catalyst for the club's surprising second-place finish.
Suarez also finished the season with a share of the European Golden Boot and was named Footballer of the Year by the Professional Footballers' Association and the Football Writers' Association.
However, his life in England has not been without controversy since he arrived from Ajax in January 2011.
He has served lengthy bans twice during his stay at Anfield, once for racially abusing Manchester United's Patrice Evra and again for biting Chelsea's Branislav Ivanovic in 2013.
Barcelona appear to be leading the race among clubs linked with the 27-year-old Uruguayan, who was expelled from the World Cup and handed a nine-match competitive international ban as well as a four-month sanction from any football activity for biting Italy defender Giorgio Chiellini.