The England international had received permission to undergo a medical this week ahead of his move across London, The Guardian said.
The report added that West Ham had accepted an offer worth 100 million pounds plus 5 million in add-ons, with the guaranteed portion of the fee expected to be paid in three instalments over two years.
Reuters has contacted both clubs for a comment.
West Ham chairman David Sullivan said last month that a gentleman's agreement meant Rice can leave despite a year remaining on his contract with the option to extend it by a further 12 months.
Rice joined West Ham's academy from Chelsea as a teenager and has been a mainstay since his breakthrough campaign in 2017-18. He has played over 200 games for the club and helped them win the Europa Conference League last month.