Antonio cost the Hammers £6 million from Nottingham Forest in August 2015 and struggled to break into Slaven Bilic's plans initially.
However, the 26-year-old soon found his feet following a run of games and ended the campaign with an impressive nine goals from 32 appearances.
The report, by Sun reporter Charlie Wyett, states that West Ham would not be keen to offload Antonio, who was plucked from non-league Tooting & Mitcham by Reading in 2008, adding that Boro are unlikely to be able to offer more than £8 million.
The Teessiders are returning to the Premier League after seven years away and spent around £15 million on ex Hammers winger Stewart Downing and former Blackburn striker Jordan Rhodes, combined, during the campaign.
Antonio, who also moved clubs in 2014, swapping Sheffield Wednesday for Forest, was discussed as a potential England call-up towards the back end of the season.