Busquets, 34, is out of contract at the end of the season and the source said Al Nassr are prepared to offer an annual salary of around €13 million net to take him to Saudi Arabia.
Major League Soccer side Inter Miami are also keen on signing the former Spain international, who said recently that he was attracted by the idea of playing in the United States.
Talks between Busquets' camp and Inter Miami have been ongoing for months, but there is still no firm agreement in place for him to move to MLS.
In the meantime, Al Nassr have made a move to sign the Barca captain as they continue to target some of the biggest names in football after announcing the arrival of Ronaldo last week.
Sources told ESPN that Busquets had considered leaving Barca this January, but he was convinced to remain at the Catalan club until the summer, when his deal expires.
Barca have not completely ruled out offering him a new deal, either, as their financial limitations make signing a replacement for first-team regular Busquets difficult. A six-month extension until next January has even been mooted.
"He has a contract until June and will stay until the end of that and then we will see," Xavi said this month in a news conference. "[It will then depend] how he ends the season, but I think it's more of a personal decision for him, based on his feelings and how he feels on the pitch.
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"For me, he is still important. We will try to convince him [to stay] if things go well, but it will depend on his personal decision. We are focused on this season for now. There is a lot to play for and we can't make plans for June when we are 100% focused on this campaign."
Busquets, who retired from international football after the World Cup, has said he would like to have clarity on his future by February. He is in his 15th season with Barca's first team and could make his 700th appearance for the club in Thursday's Supercopa semifinal against Real Betis -- only Lionel Messi (778) and Xavi (767) have made more appearances.
LaLiga sides are allowed three non-EU players -- there is an agreement which allows African players to play regardless -- in their squad. Brazilian forward Raphinha has an Italian passport, so Araujo was Barca's only non-EU player. They now have all three spots clear.