Sky Sports report that the most lucrative offer comes from the MLS Club, part-owned by David Beckham, who want to give him a three-and-a-half year contract.
Arsenal are also said to be monitoring his situation.
The 31-year-old, who has won two Premier Leagues and the Europa League since joining the London club in 2013, will reportedly not make any decision on his future until after Chelsea's FA Cup final against the Gunners on Saturday evening.
THE EXPERT VIEW
Eurosport Spain's Jose Arronis: Every summer there appears to be rumours between Willian and Barca but this could be the year.
Willian would have to accept a secondary role at Barcelona because the Lionel Messi, Luis Suarez and Antonine Griezmann trio is non-negotiable.
Ansu Fati is a future star and next season he will be a more important member of the team.
Real Madrid chase 'next Raphael Varane' - Euro Papers