Xavi, who is currently in charge of Qatari team Al-Sadd, held talks with Barca’s sporting director Eric Abidal and chief executive Oscar Grau on Friday and Saturday.
But a source has told AFP Sport that Grau and Abidal were returning to Barcelona on Sunday, after Xavi said he would not be giving his agreement straight away.
Xavi could yet resume negotiations in the coming days or, more likely, in the summer once the season has finished.
In the meantime, Valverde remains in charge and the plan is that he will take training as usual on Monday morning.
Yet it remains to be seen how he responds, with the current Barca coach likely to have been unimpressed by the club’s indiscreet pursuit of Xavi.
Abidal and Grau met Xavi on Friday in Qatar and then again on Saturday night following Al-Sadd’s Qatar Cup semi-final win over Al-Rayyan.
The suggestion on Saturday had been that Xavi would make a decision as early as Sunday but the club’s iconic former midfielder wants more time.
– Tough decision –
His situation is complicated by the fact he would like to lead Al-Sadd in the Qatar Cup final, which will be played on January 17.
It would also be a daunting task to join Barcelona mid-season. The team are level on points in the league with Real Madrid and face a tough last-16 tie in the Champions League next month against Napoli.
There is also the club’s political situation to consider, with Xavi believed to be key to the hopes of presidential candidate Victor Font, ahead of the club elections due next year.
“It’s a difficult decision,” wrote El Pais on Sunday. “If he accepts, he is put under the wing of a weak leadership, with elections due in 2021. But if he says no, he may be criticised for refusing to return home.”
Xavi is also said to want to show respect to Valverde, who has overseen Barcelona winning two La Liga titles in his two seasons in charge.
But Champions League collapses against Roma and Liverpool have not been forgotten and poor performances in recent weeks have prompted fears of a repeat.
Barca were beaten 3-2 by Atletico Madrid on Thursday in the Spanish Super Cup in Saudi Arabia.
“I respect Barcelona, I respect Valverde and I respect my club and I’m doing my job here,” Xavi said on Saturday.
“I’m focussing on Al-Sadd. Yes Abidal is my friend, I met him many times,” he continued. “They were here to talk to me and we discussed many things. And they were here to check on Ousmane Dembele.
“I cannot inform you too much, just that I met Abidal and he is my friend.”
But Xavi added: “I cannot hide that it’s my dream to coach Barcelona, I said it many times.”
French international and Barcelona forward Dembele has been in Qatar for treatment on a hamstring injury.
The 39-year-old Xavi, who won the World Cup with Spain in 2010, came through Barcelona’s academy and played 855 senior games for the club.
After an emotional farewell, he joined Al-Sadd as a player in 2015 before being appointed coach last July.
Asked about talks between Barca and Xavi, Al-Sadd’s sports director Muhammad Ghulam Al-Balushi told the Qatar-based BeIN broadcaster: “I will not deny that. There are negotiations with Xavi and everyone is talking about them… but I can say that Xavi so far is at Al-Sadd.
“The final decision is in the hands of Al-Sadd’s management, in the hands of the manager and the hands of Barcelona’s management.”