Speculation surrounding the Catalan club's interest in the Uruguay international, currently banned for four months for biting Italy's Giorgio Chiellini at the World Cup, has been rife for several weeks.
However, following the 27-year-old striker's apology - in which Barca are suspected to have had a major influence in procuring - the La Liga side have stepped up their pursuit.
On Tuesday sporting director Andoni Zubizarreta spoke in complimentary tones, suggesting "he has the necessary quality to play for Barca", and the club have followed that up with a request for a meeting with Ayre.
Ayre is expected to meet Barcelona officials in London today to listen to their proposals.
However, there is likely to be plenty of negotiation to follow as Liverpool will not even consider doing business unless Suarez's high-value buy-out clause - possibly as much as £80million - is met as they do not believe the four-month suspension has reduced the value of a player who was the Barclays Premier League's top scorer last season with 31 goals.
The confidential clause was inserted into the new lucrative £200,000-a-week contract he signed in December but it is reported Barcelona will try to secure the player for between £50million and £60million.
As an incentive they may offer forward Alexis Sanchez, valued at £30million, as a makeweight in the deal as the Reds are known to be interested in the player, but the Chile international is also a target for other big clubs around Europe.
Last summer Arsenal failed in their attempts to sign Suarez after a bid of £40,000,001 was made in the mistaken assumption that would trigger his release when it actually only meant that Liverpool had to inform their striker of the Gunners' offer.
Since then Liverpool have moved to make Suarez's contract situation more definitive in terms of the price of his release.
At a press conference on Wednesday, Barcelona president Josep Maria Bartomeu refused to confirm that talks had opened over Suarez, but hinted that the controversial striker was on the club's agenda.
Bartomeu refused to speak directly about his team's interest in Suarez, but followed Zubizarreta's lead in commending the Uruguayan for his apology to Chiellini.
"I announced months ago that the team would undergo a deep renovation but we cannot reveal details because we don't want to give clues away to any of our rivals," Bartomeu said.
"Our representatives are constantly travelling to other countries because of potential signing.
"Suarez is a Liverpool player so I can't talk about him, he belongs to another team, a rival team. But we are all football men, and saying sorry is honourable, it helps the competition.
"He did something that wasn't right, so it's the responsibility of everyone in football, be it Liverpool or anywhere else, to remember that he has said sorry. Admitting you have done something wrong is very important."
Suarez's ban means that he will not be able to play until October, whether that be in Barcelona, Liverpool or elsewhere.
"Luis has apologised and that is honourable, he's taken a step towards rehabilitating, the football world should support him and help him," added Bartomeu.
"As a football fan I hope he can turn a corner."