Carrick is currently playing for the Hammer's upcoming FA Cup tie opponents Manchester United, but his future is unclear with his current deal set to expire in June.
The 34-year-old made his name at West Ham before leaving to join Tottenham in 2004, and then signing for United for £18 million in 2006.
He has gone onto win five league titles with United and one Champions League trophy.
The Sun's report states West Ham have approached the experienced midfielder and are planning on offering him a one-year deal.
The ex-Hammer may decide he wants a deal which is more long term, but the prospect of rejoining his former club has to be tempting with them set to move into their new stadium next season.
West Ham are only narrowly behind Manchester United in the table, and Carrick's versatility to also play in defence if required makes him an excellent acquisition on a free transfer if a deal can be struck.