Xavi is currently in charge of Qatari team Al Sadd, which confirmed on Saturday he has held talks with Barca, AFP reported.
The Spaniard who once pulled the strings at the heart of Barcelona's midfield in its golden era admitted it is his "dream job" to take charge of the Catalans.
Barcelona's sporting director Eric Abidal and chief executive Oscar Grau met Xavi on Friday in Qatar and then again on Saturday night following Al Sadd's Qatar Cup semifinal win over Al Rayyan.
According to Spanish sports daily Mundo Deportivo, Abidal and Grau have offered Xavi the chance to replace Ernesto Valverde immediately, even if Xavi's preference would be to wait until the end of the season.
"The preferred thinking of Xavi would be to arrive at Camp Nou in the summer, a message he has transmitted," Mundo reported on Sunday. "Negotiations remain open and the possibility has not been discarded of him signing imminently."
Another Spanish sports daily Diario AS reported that Xavi will take his time to decide, while noting that with "every passing minute it seems to become more difficult for Xavi to say no to the offer of joining immediately."
Xavi's 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 midseason. The team is level on points in the league with Real Madrid and faces 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 criticized 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 was 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 focusing 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."