Ronald Koeman, Roberto Martínez and Thierry Henry have also been considered for Barcelona job but Xavi seems to be the leading candidate should his club Al-Sadd leave him for fresh challenges.
World Cup victor Xavi came through the Catalans' academy and played 855 senior games for the Camp Nou side.
According to Spanish outlet AS, sporting director Eric Abidal and CEO Oscar Grau flew to Doha for a two hour meeting with Xavi in which they offered the 39-year-old the chance to take over from Valverde for next season.
Qatari side Al Sadd say they have not received an approach from Spanish champions Barcelona for manager Xavi. I've met him many times, so I can not say anything, just that I respect Barcelona, I respect Valverde, I respect my club.
"I was with Abidal".
The former Spain worldwide is taking his time to decide, the report highlights, not helped by a busy schedule with his side, Al-Sadd.
"But for now I'm focused on Al-Sadd".
Xavi spent a staggering 17 years of his career at the Nou Camp before moving to Al Sadd in 2015.
" I cannot hide it's my desire to trainer Barcelona, I've claimed it often times in lots of meetings, every person recognizes I sustain Barcelona from all-time low of my heart. Sorry I can not give any more information", Xavi informed a press conference after his side beat Al Rayyan on Saturday to get to the last of the Prince's Cup.
"Xavi is negotiating with Barcelona", the sporting director said in quotes reported by Marca.