The midfielder's current deal at Camp Nou is due to expire next June, and there remains widespread speculation surrounding his future with the Catalan outfit.
Busquets is believed to be tempted by a move to the MLS, but a recent report claimed that the former Spain captain would be willing to sign a new contract with Xavi's side under the right circumstances.
According to journalist Toni Juanmarti, Barcelona believe that offering a six-month deal until December 2023 would suit all parties.
The report claims that Busquets had initially been looking to leave Camp Nou in January in order to undergo pre-season with his new MLS club before the start of the campaign in February.
However, Barcelona are not prepared to let the experienced midfielder depart at the start of 2023, so he could sign a new deal until next December with a view to moving to a North American side ahead of the 2024 campaign.
Inter Miami are said to be firmly at the head of the queue for Busquets, who announced his retirement from Spain duty following the 2022 World Cup.
Barcelona boss Xavi is said to be desperate to keep hold of his former teammate, as he remains a key player for the club, featuring on 16 occasions in all competitions this term.
Busquets made his first-team debut for the Catalan side in 2008, and he has featured on 696 occasions for the club in all competitions, contributing 18 goals and 42 assists.
The midfielder recently said that he was looking to make a decision on his future "around February".
"It's my last year, I know a lot has been said, but I've always spoken in the same way. I have nothing signed from one side or the other. We'll see how it goes, but around February or so I would like to be clear and decide," he told El Larguero.
Meanwhile, Barcelona sporting director Jordi Cruyff recently hinted that the club want Busquets to stay.
"He is a super special player, a 'one club man', who has spent his whole life in a club and there comes a time when, with such a special player, the first thing you have is respect for the how and when of the decision you want to make," Cruyff told Mundo Deportivo.
"Obviously, the club is in permanent contact with him and with his agents, which is normal, and when the time comes, a joint decision will be made. But the club does not have the decision, the player has the right to decide and also the contract ends."
Barcelona will be back in competitive action against Espanyol on December 31.